A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing


  • 5 years ago

Use Lagrange multipliers to find the maximum or minimum values of f(x,y) subject to the constraint. f(x,y)=3x-2y, x^2+2y^2=99

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

    Solution 1 (without multipliers): If you look at f you can see that its graph is a plane through the origin and grows with the highest rate in the direction of the gradient vector\[\nabla f(x,y)=(3,-2)\] so you just have to go to the constraint (which is an origin centered circle with radius square root of 99) in that direction. This right point to the maximum is \[\sqrt{99}(\frac{3}{\sqrt{13}},\frac{-2}{\sqrt{13}})\] where the value of f is \[\sqrt{99}(\frac{9}{\sqrt{13}}+\frac{4}{\sqrt{13}})=\sqrt{99\cdot 13}\approx 35.87\] By symmetry you get the minimum is at point \[\sqrt{99}(\frac{-3}{\sqrt{13}},\frac{2}{\sqrt{13}})\] where f is equal to -35.87. Solution 2 (using Lagrange multipliers):The function is \[f(x,y,)=3x-2y\] and the constraint is \[g(x,y)=x^2+y^2=99.\] Define the auxiliary function \[\Lambda(x,y,\lambda)=f(x,y)+\lambda(g(x,y)-99)=3x-2y+\lambda(x^2+y^2-99).\] The system of linear equation formed by the partial derivatives of Lambda: \[\begin{array}{rcl} \frac{\partial\Lambda}{\partial x}=3+2\lambda x&=&0\\ \frac{\partial\Lambda}{\partial y}=-2+2\lambda y&=&0\\ \frac{\partial\Lambda}{\partial \lambda}=x^2+y^2-99&=&0. \end{array}\] It has the solution: \[x=-3\frac{\sqrt{99}}{\sqrt{13}},\quad y=2\frac{\sqrt{99}}{\sqrt{13}},\quad \lambda=\frac{1}{2}\frac{\sqrt{13}}{\sqrt{99}}\] and \[x=3\frac{\sqrt{99}}{\sqrt{13}},\quad y=-2\frac{\sqrt{99}}{\sqrt{13}},\quad \lambda=-\frac{1}{2}\frac{\sqrt{13}}{\sqrt{99}}\] The first one gives the minimum and the second one the gives the maximum of the function subject to the given constraint.

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

    Oh... now i see it's 2y^2 at the constraint, so no big deal you should do almost the same but now solve the system of equation 3+2λx=0 −2+4λy=0 x^2+2y^2−99=0.

  3. 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...


  • 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.