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anonymous

  • one year ago

(a) Find the critical numbers of the function f(x) = x^8(x − 3)^7. I dont have examples like this problem so Im having a bit of trouble, please help!

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  1. idku
    • one year ago
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    what is the interval?

  2. idku
    • one year ago
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    is there an interval on which you are to find critical numbers, or there is no interval given >

  3. idku
    • one year ago
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    anyway, I will tell you what to do, before you get to answer me.

  4. anonymous
    • one year ago
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    the interval is not given

  5. idku
    • one year ago
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    So, you have to: 1) Differentiate the function (take the derivative of the f(x) ). 2) Set f`(x)=0, and the x values that are solutions to this are critical numbers. 3) Critical numbers are also values where f`(x) is undefined - IFF f(x) is defined on that value(s) - but if f(x) is undefined at this value(s) then it is NOT a critical number.

  6. idku
    • one year ago
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    Also, if you were given the interval [a,b] (specifically a closed interval) then a and b would automatically be (also) your critical numbers.

  7. idku
    • one year ago
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    Is the outline/plan clear ?

  8. anonymous
    • one year ago
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    yes! thank you!

  9. idku
    • one year ago
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    ok, can you find f`(x) for me pliz ?

  10. idku
    • one year ago
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    you can use product rule (or logarithmic differentiation)

  11. anonymous
    • one year ago
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    working it out now

  12. idku
    • one year ago
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    Product d/dx [ f × g] = f` g + g` h

  13. idku
    • one year ago
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    ok, take your time

  14. idku
    • one year ago
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    i will post the provwe for product rule now.

  15. anonymous
    • one year ago
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    so i have 8x^7 (x-2)^7 what do i do now?

  16. idku
    • one year ago
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    y = f * g (where f and g are some functions of x) use logarithmic diff. ln(y) = ln(f * g) ln(y) = ln(f) + ln(g) now, f and g are big function, so for any function H, there is a chain rule such that: d/dx [ ln( H(x) ) ] = ( 1 / H(x) ) * H`(x) So here we will use chain for f and g differentiating: (dy/dx) * y=f`/f + g`/g dy/dx = (f`/f + g`/g) * y then look at what y is from the beginning of the problem and substitute. find common denom, and cancel things out

  17. idku
    • one year ago
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    No, that is not correct. don't think so

  18. idku
    • one year ago
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    f(x) = x^8(x − 3)^7 f`(x) = (x^8)` * (x − 3)^7 + x^8 * ( (x-3)^7 )`

  19. idku
    • one year ago
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    f`(x) = 8x^7 * (x-3)^7 + x^8 * 7(x-3)^6

  20. idku
    • one year ago
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    anyway, u got f`(x) then find x values for f`(x)=0 f`(x)=undefined i got to go

  21. anonymous
    • one year ago
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    ok, thank you!

  22. anonymous
    • one year ago
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    I got x=0,3 and 24/15

  23. anonymous
    • one year ago
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    (b) What does the Second Derivative Test tell you about the behavior of f at these critical numbers? At x=? the function has a local max, a local min or neither?

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