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Stuck with Optimisation question, help?

  1. Jul 3, 2013 #1
    S=8x2ln(1/2x)

    Find the value of x that gives a maximum.

    So far I have got, by differentiating: x2+ln(1/2x). [could be wrong]

    Btw the way in the question x is a ratio and so cannot equal zero.

    Please help and explain how to do it thanks :)
     
  2. jcsd
  3. Jul 3, 2013 #2

    SteamKing

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    First, brush up on taking derivatives. What you have in the OP is way wrong.
     
  4. Jul 3, 2013 #3
    Can you show the steps you took when you differentiated the function?
     
  5. Jul 3, 2013 #4
    I'll take you through my working:

    [itex]\frac{dS}{dx}[/itex]=16x3+16xln(1/2x)

    16x3+16xln(1/2x)=0. For maximum

    16x(x2+ln(1/2x))=0

    X2+ln(1/2x)=0
     
  6. Jul 3, 2013 #5
    I'll take you through my working:

    [itex]\frac{dS}{dx}[/itex]=16x3+16xln(1/2x)

    16x3+16xln(1/2x)=0. For maximum

    16x(x2+ln(1/2x))=0

    X2+ln(1/2x)=0
     
  7. Jul 3, 2013 #6

    SteamKing

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    I'm confused. In the OP, S = 8x^2 * ln(1/2x)

    In post #4, dS/dx = 16x^3 + 16x*ln(1/2x)

    ????

    I think it would be better if you post the entire problem from the beginning and don't try making shortcuts.
     
  8. Jul 3, 2013 #7
    yeah the power rule bud, did you use it right?
     
  9. Jul 3, 2013 #8
    I was using the product rule: [itex]\frac{d}{dX}[/itex](fg)=f[itex]^{|}[/itex]g+fg[itex]^{|}[/itex]
     
  10. Jul 3, 2013 #9
    okay so the first part doesn't vanish they both have x terms so it seems like you lost a a term :/ the natural log term
     
  11. Jul 3, 2013 #10
    Confused :confused: can someone write out the correct solution?
     
  12. Jul 3, 2013 #11
    Is that the original problem?
     
  13. Jul 3, 2013 #12

    SteamKing

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    Sorry, we don't do that here at PF.

    Look, you don't have a problem with optimization. You have a more basic problem with understanding how to take a derivative.
     
  14. Jul 3, 2013 #13
    I don't believe I have a problem with basic differentiation as I have studied it for well over a year and have been fine with it, however I am definitely not an expert. Where about do you believe the problem lies?
     
  15. Jul 3, 2013 #14
    I think I could have the answer if someone just helps me find x from: (x+ ln(1/2x))=0
     
  16. Jul 3, 2013 #15
    Is this the original question?
     
  17. Jul 3, 2013 #16
    Original Question:

    A communications cable has a copper core with a concentric sheath of insulating material. If x is the ratio of the radius of the core to the thickness of the insulating sheath, the speed of a signal along the cable is given by:

    S=8x2ln([itex]\frac{1}{2x}[/itex])

    Find the value of x that gives the maximum speed.
     
  18. Jul 3, 2013 #17
    okay. ## S_{x}## is what you need that is the partial derivative of S with respect to x, so you get something that looks like this after taking the first derivative ##S_{x} = -16x ln( {2x} ) - 4x## So what I did was product rule then chain rule, I am not exactly sure if the last part with subtraction is right I am a little rusty with natural logs so that means that ##dS = S_{x} dx## that should be what you need. But there is still the problem of max. and what you can do to find the max is second derivative test.... I think I am missing one term but I am not sure...
     
    Last edited: Jul 3, 2013
  19. Jul 3, 2013 #18

    micromass

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    Partial derivatives?? Why do you need them here. This is a single variable problem. Please don't confuse the OP by bringing up multivariable calculus.
     
  20. Jul 3, 2013 #19

    D H

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    Tenshou: Don't help if you can't solve the problem.
     
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