A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • 5 years ago

Using Lagrange multipliers to find the maxium and minimum values of the function f(x,y,z)+xy^2z^3 subject to x^2+y^2+2z^2=25. PLEASE HELP

  • This Question is Closed
  1. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    So you're gonna go through the equation taking derivatives with respect to all three variables but one at a time. So first, derive the equation with respect to the variable 'x' while treating y,z like constants, that's one derivative. Then do the same again but with only y as the variable, and then again with z.

  2. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Let me know when you have done that much!

  3. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so I need to take partials in respect to each variable fx, fy, fz?

  4. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so fx=y^2z^3 fy=2xyz^3 fz=3xy^2z^2

  5. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes, partials. gimme a second

  6. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    wait a second, rephrase your question. what do you mean by f(x,y,z)+xy^2z^3 ??

  7. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    do you have two functions?

  8. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    f(x,y,z)=xy^2z^3 subject to the constraint of x^2+y^2+2z^2=25

  9. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I'm in the middle of two other quesitons. If you're not rushed, I can get back to you on this one. alright?

  10. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    not rushed it is due in the morning, thank you for your help. this site is fun

  11. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    brb

  12. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    well, in 30

  13. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    okay

  14. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i couldn't get thru, som' about about the server not receiving my post but I'm still looking at the problem. Just grabbed my snack and I'm not browsing my calc book.

  15. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    thanks

  16. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    it's an optimization. very much in the line of those classic optimization problems in simpler calculus but I haven't done one in a bit

  17. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    made any progress?

  18. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i am confused because i left my book at home and I am at school

  19. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    something to do with taking the gradient of the constraint

  20. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    we have not done this type of problem, and i have no information in my notes

  21. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    which calculus are you taking?

  22. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    calc 4, 3-d vectors, multivarible

  23. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    nice

  24. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    problem is only worth 5 points but i think it is important to know how to do it

  25. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i agree

  26. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i'm dining and browsing pages at the same time...

  27. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    in a weird way, i miss school; funny how that works

  28. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    eat, i dont want to ruin your dinner, just get back to me when you are finished

  29. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    don't worry about it

  30. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i only have 2 more math classes and 3 physical chem classes and 3 chem engineering classes and then I am done

  31. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    lol, then heading for the petroleum industry?

  32. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i want to do desalinization, water filtration, water is the next oil!

  33. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    or if i could figure out how to make coal a clean fuel source

  34. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the chapter on lagrange multipliers starts: in my applications we must find the extrema of a function of several variables when the variables are restricted in some manner.

  35. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    gimme a quick second

  36. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    f (xyz) =xyz + (2xz + 4yz + 8xy − C) does this formula seem familiar

  37. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    why?

  38. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    you're end up with 2 numbers at the end, the maximum and minimum.

  39. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    correct this is what i need

  40. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i need to solve for x and y

  41. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    correct? i could be wrong

  42. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    actually, it's a little different

  43. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    z is negilble because it is the depth and not going to be a min or a maxium

  44. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the minimum and maximum values will be points on the graph which happens to be a 3 variable function

  45. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    all the variables are going to count

  46. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    okay

  47. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the maximum and minimum values will be points in a 3 dimensional space, so they'll look like (x1,y1,z1) and (x2,y2,z2)

  48. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    if we didn't have the constraint, it would be straight forward. we'd find the partials, set them all equal to zero and then solve for the three variables.

  49. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    since we would have fx=0, fy=0, fz=0; all equations in their own right, and we have 3 variables, we would be able to solve this problem completely.

  50. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    depending on how many CRITICAL points we find, some could be maximum(s) and others minimum(s)

  51. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    thoughts?

  52. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so take the partials and set them equal to zero and solve for a x, y, or z and plug the values into the equations

  53. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    is that the first step?

  54. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    that would only be if we didn't have a constraint; since we have a constraint, we need to find a way to include it in our optimization.

  55. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so take the constraint and solve for one of the variables. pick which ever one you'd like

  56. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    by the way, i take it you're in college...what year?

  57. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i am a junior

  58. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    where?

  59. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so the contratint is x^2+y^2+2z^2=25

  60. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    and let me know once you have solved for one of the variables in the constraint...

  61. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    at Ohio State

  62. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    that's correct. chose one variable, x, y or z and solve for it

  63. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    good stuff

  64. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I graduated from Stanford and I'm currently working in the DC area

  65. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    chose a variable that's gonna make it easy on you cuz we'll have to plug whatever that variable is equal to into the f(x,y,z) function!

  66. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    once we do that, we'll have one function with only 2 variables and we can find the two partials, set them equal to zero and solve!

  67. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    show me your work and let me know what you're thinking so that I can guide you in case you get lost...

  68. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    take your time

  69. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    with the 2 present in the constraint the value i get solving for x=5-y-\sqrt{2z ^{2}}\]

  70. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[x=5-y-\sqrt{2z ^{2}}\]

  71. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    thank you! much better

  72. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    let me check...

  73. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    walk me thru how you got this step by step...or just to the part before you took the square root

  74. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the constraint equation is \[x ^{2}+y ^{2}+2z ^{2}=25\]

  75. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    solve for X, so I squared both sides

  76. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    first you took y^2 and 2z^2 to the other side

  77. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yeah, then squared both sides

  78. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    and then you had \[x^2 = 25 - 2z^2 - y^2\]

  79. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yep

  80. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the square of the right isn't exactly what you got bradley

  81. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so should i not square the equation?

  82. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    no, that was right

  83. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    okay so what do i do with it? do i plug it into the partial?

  84. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the square root of 25-2z^2-y^2 doesn't simplify to that.

  85. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    dude, give 10 minutes

  86. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    okay you are fine

  87. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh how i hate contraints

  88. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes it does simplft to that

  89. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i hate this problem

  90. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i'm back and we're gonna solve this problem. sorry for the delay

  91. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i need to find a gradient of the the f(x,y,z) right?

  92. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes, but we need to take the constraint into account

  93. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so we solve for one variable, x like you did above

  94. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the \[x= \sqrt{25-2z^2-y^2} \neq 5 - y - \sqrt{2x2}\]

  95. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the \[x=\sqrt{25-2z^2-y^2} \neq 5 - y - \sqrt{2x^2}\]

  96. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    are we good here?

  97. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes

  98. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    if you're not sure why those two are NOT equal, look it up when you get a chance.

  99. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i figured that part out

  100. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    now back to our function, plug this new found value of x into our original function replacing it at every place where x shows up

  101. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    okay i already did that

  102. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so the original function was \[f(x,y,z) = xy^2z^3\]

  103. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    what did you get?

  104. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    here's another thing to think about: if you had chosen y instead of x, we would be square it and getting rid of the nasty square root. you wanna do that again?

  105. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    do let me know what you're thinking and if you're start getting lost...

  106. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\sqrt{25-2z ^{2}-y ^{2}}y ^{2}z ^{3}\]

  107. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh did not think of that

  108. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    but you see why that would have been smart?

  109. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    still is...

  110. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    why does the y solve and the radical go away?

  111. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh never mind i see it, let me check

  112. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Bradley, we're solving this without using Lagrange Multipliers (LMs)....this is gonna give us another but apparently the LMs provides a nicer way....we're gonna look into that after.

  113. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    it's very good to see you check your work and see why things are why they are. keep it up!

  114. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    once you have the function of \[f(x,z)\] you can find fx and fz, set them equal to zero and solve them. you'll have 2 equations and 2 variables. you see why it's gonna fx and fz? because why will have been replaced by those two.

  115. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[x(25-2z ^{2}-x ^{2})z ^{3}\]

  116. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    what is this? always remember to represent functions in terms of f(x)=something...

  117. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    in this case it would be \[f(x)=....\]. multiply it out and then find fx and then fz.

  118. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so what i wrote before is the f(x,z)

  119. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    that's correct

  120. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    now can you find fx and fz, set them equal to zero and solve for x and z?

  121. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so take the partial of each?

  122. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    find each partial

  123. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    note: this will give you the answer, but not using the method the suggested. i'm looking at exactly how Lagrange Multipliers work. it's something like \[\Delta f(x,y,z) = \lambda \Delta g(x,y,z)\] where \[g(x,y,z)\] will be your constraint function.

  124. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    partial of fx=\[25z ^{3}-2z ^{5}x-x ^{3}z ^{3}\]

  125. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the Lagrange multiplier is no jok but we might be able to figure out if you stick around :)

  126. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    fz=\[75xz ^{2}-10z ^{4}x-3x ^{3}z ^{2}\]

  127. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i am here until it is done

  128. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    damn, these are some nasty looking numbers!

  129. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    nasty looking functions i should say

  130. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yep, i think this is why the lagrange multiplers are needed

  131. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    absolutely right

  132. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    any way, let's finish this first. if we can, lol

  133. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so we set those partials to zero and see what happens

  134. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    are you good with algebra?

  135. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    pretty good

  136. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    that's what I like to hear! so let's simply the hell out of these two and solve the system of 2 equations to find the solution

  137. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so set them both equal to zero, simplify, and then solve for x and z

  138. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    let me know if you get stuck

  139. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i see alot of quadratic forumla in my future

  140. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    lol

  141. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    we might break half way and make the jump to langrage...

  142. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    there are multiple variables so i can do the quadatic, even after factoring out some things

  143. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    do you think it would be better to take the graident of f(x,z)

  144. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    might be a little easier but not much easier. i'm looking at a \[5x^3-3x^2-50\] for this one

  145. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    a \[5x^3-3x^2-50=0\]

  146. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    a calculator might solve it...

  147. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    where does that come from?

  148. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[f _{x}=25z^3-2z^5x-x^3z^3=0 => f_{x}=25-2z^2x-x^3\]

  149. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yeah i see that

  150. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[f_{z}=75xz^2-10z^4x-3x^3z^2=0 => f_{z}=75-10z^2-3x^2\]

  151. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the last part was \[-3x^2\]

  152. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    if you solve that, eliminate \[z\] by multiplying the first with -5 and adding both equations you get what i got.

  153. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    check my work, i may have made an algebraic error while I double check this langrage business

  154. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    at the very least, you'll have an answer for this question and then we can worry about the method.

  155. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i am not seeing the x^3 is coming from

  156. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    in which partial?

  157. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    x or z?

  158. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    no when i added the 2 i got 12x^2 - 50

  159. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    in the fx why did you not factor out z^3

  160. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I did. look at the part after the => sign

  161. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    am I missing something here? plz do let me know, I could have easily made a mistake...

  162. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    btw, I admire your persistence. I know you could say the same for me but I enjoy this kind of thing. Usually I enjoy working on challenging and interesting problems, but I didn't when I was a student. I did, but I didn't put in the work. so good job thus far.

  163. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    by the way, I have a problem here in my book which is a superb example for solving this problem with lagrange multipliers. I don't have a scan, but I can take a pic with my camera. so if you'd like you can send your email so i can forward it to you.

  164. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    lets start with the basics f(x,z)=\[25xz^3-2z^5x-x^3z^3\]

  165. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    brower.brad@gmail.com

  166. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    fx=\[25z^3-2z^5-3x^2z^3\]

  167. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    fx=\[z^3(25-2z^2-3x^2)\]

  168. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    btw, what you typed in earlier: f (xyz) =xyz + (2xz + 4yz + 8xy − C) does this formula seem familiar. this is what we'll end up referring back to for the langrage.

  169. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    was that in your notes somewhere?

  170. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    fz=\[75xz^2-10z^4x-3x^3z^2\]

  171. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    that was from a volume equation the variables were off, dont think it matters

  172. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    fz=\[xz^2(75-10z^2-3x^2)\]

  173. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i see the problem in the book but i guess i am having trouble seeing what to do with what i find

  174. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so you have the example to a similar problem as well that is being solved using Lagrange Multipliers?

  175. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    by the way, the x^3 came from the original fx that you gave me. I didn't check it. if it was true, that's where the x^3 was coming from.

  176. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    must have been a type, there shouldn't be a x^3. just checked.

  177. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yea, so it should nicely reduce then

  178. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    okay

  179. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so i got 12x^2-50

  180. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so then it is x=\[\sqrt{50}/12\]

  181. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[5\sqrt{2}/12\]

  182. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    what about z?

  183. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    by adding-5 to fx and adding to fz the z^2 canceled

  184. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    right, but now you plug the value of x into one of them to solve for z as well. remember?

  185. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    sorrying multiply -5 to fx

  186. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so plug in the \[5\sqrt{2}/12\] to the orignal functions

  187. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    and solve for z?

  188. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    right

  189. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    this number is nasty

  190. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    it shouldn't be

  191. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    maybe i am doing poor algebra, wait do i plug it into f(x,y,z) or f(x,z)

  192. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    into f(x,z)

  193. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[f_{x}=25-2z^2-3x^2\] and \[f_{z}=75-10z^2-3x^2\]

  194. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\sqrt{50}/12(25-2z^2-(\sqrt{50}/12)^2)z^3\]

  195. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    subtract those two: \[f_{z}-f_{x} = 50-8z^2\]

  196. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    haha so much easier

  197. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    5/2

  198. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    that's right. now plug that into either fx or fz

  199. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so if i plugged the z value into fx, i would solve for X?

  200. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    right

  201. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    what do you have for me?

  202. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    you realize that once you get the x we'll now have x and z, and remember the expression of y you got earlier by solving for y? well, if you go and plug this x and z in it, you'll have the y which will give you a point in 3D (x,y,z) and that will be the solution!

  203. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I just got through looking at the Lagrange and it is considerably easier once you get it. let me know when you're ready to move on.

  204. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the x is difficult because of the fractions

  205. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    It's nearing 11 o'clock my time, so we'll have to hurry up a bit :)

  206. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    awww, not a fan of fractions are ya? no worries

  207. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    just teasin' by the way,

  208. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i agree i am about done

  209. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so let's figure out this fraction business

  210. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[f_{x,z=5/2}=25-2(5/2)^2-3x^2x\]

  211. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    simplying gives us: \[0=25-25/2-3x^2=50/2-25/2-3x^3\]

  212. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    50/2-25/2=25/2, right?

  213. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    because you get (50-25)/2=25/2

  214. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so we now have \[25/2-3x^2=0 => 25/2=3x^2\]

  215. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    you can get x from there

  216. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\{5}/6\]

  217. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    5/6

  218. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    not exactly...

  219. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    sorry my brain is fried the whole thing must be squared

  220. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    exactly

  221. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    don't worry, that makes two of us

  222. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\sqrt{25/6}\]

  223. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    which simplifies to...

  224. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[5/\sqrt{6}\]

  225. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes, but by convention you don't leave a radical in the bottom so you...

  226. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    multiply both top and both by the same radical and you get...

  227. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[5\sqrt{6}/6\]

  228. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    that's it

  229. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    now, if we didn't make any stupid mistakes along the way, those two should be two of the 3 you need to make a point in 3D

  230. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    now plug all three into the equation

  231. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    now plug those into the y expression you got earlier and you should get the third point to form (x,y,z)

  232. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    once you have the point, that's it. that should be the optimal point you were looking for to satisfy whatever the problem was asking you to satisfy in real life

  233. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    now onto lagrange: given \[f(x,y,z)\], and the constraint \[x^2+y^2+2z^2=25\], let \[g(x,y,z)=x^2+y^2+2z^2-25\]. Now, all the wise Lagrange himself said was that given that the gradient of \[g(x,y,z\] neve equals zero, there exist a constant such that the gradient of \[f(x,y,z)\] equals the constant times the gradient of \[g(x,y,z)\].

  234. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    okay

  235. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    first, you know now how you're gonna get the y?

  236. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    and at least have your optimal point? before we do this now. we're just gonna quickly set up the equations and I'm gonna hit the pellets

  237. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    *sheets

  238. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yeah y=\[\sqrt{25/3}\]

  239. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    or \[5\sqrt{3}/3\]

  240. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    sweet. i haven't checked it. make sure all the algebra's right.

  241. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    go ahead and go to sleep i appreciate all your help, it was above the call of duty

  242. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    do you see how I wrote the constraint as the function g(x,y,z) ?

  243. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes

  244. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so langrage gives you a bunch of easy equations to solve: here they are:

  245. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    he says that there exists a constant, \[lambda\] that when multiplied by the constraint, you get the solution. interesting man

  246. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    okay

  247. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so here are the equations: fx = lambda*gx, fy = lambda*gy, fz = lambda*gz, and finally, we have our constraint: g(x)=0;

  248. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    lambda as I am using it here is of course \[\lambda\]

  249. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    with the g(x,y,z)=0 we have 4 equations which is great because we have 4 variables: x,y,z, and \[\lambda\]

  250. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so take the partial of gx, gy, gz and solve for lambda

  251. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    that is exactly correct! of course you'll also be taking partials of fx,fy,and fz but remember that here we're talking about the original \[f(x,y,z)=xy^2z3\]

  252. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes! awesome, i got it! thank you so much for your help

  253. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    in the process you'll solve for the lambda, x,y, and z. you're interested in the x,y,and z and if we did everything right, they should match what we got earlier.

  254. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i understand it know thanks so much

  255. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    good to know where everything comes from

  256. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    or that there are easier ways of getting things done :)

  257. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    if i ever get stuck can i email you?

  258. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    you're welcome. you have my email, shoot me one if you ever need some more help. i don't visit here often.

  259. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    you beat me to it; absolutely.

  260. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    thank you very much

  261. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    aite man, ciao

  262. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    sleep well

  263. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    thank you

  264. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.