Homework Help: Stuck with Optimisation question, help?

1. Jul 3, 2013

CallumC

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.

2. Jul 3, 2013

SteamKing

Staff Emeritus
First, brush up on taking derivatives. What you have in the OP is way wrong.

3. Jul 3, 2013

Number Nine

Can you show the steps you took when you differentiated the function?

4. Jul 3, 2013

CallumC

I'll take you through my working:

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

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

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

X2+ln(1/2x)=0

5. Jul 3, 2013

CallumC

I'll take you through my working:

$\frac{dS}{dx}$=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

SteamKing

Staff Emeritus
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.

7. Jul 3, 2013

Tenshou

yeah the power rule bud, did you use it right?

8. Jul 3, 2013

CallumC

I was using the product rule: $\frac{d}{dX}$(fg)=f$^{|}$g+fg$^{|}$

9. Jul 3, 2013

Tenshou

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

10. Jul 3, 2013

CallumC

Confused can someone write out the correct solution?

11. Jul 3, 2013

Tenshou

Is that the original problem?

12. Jul 3, 2013

SteamKing

Staff Emeritus
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.

13. Jul 3, 2013

CallumC

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?

14. Jul 3, 2013

CallumC

I think I could have the answer if someone just helps me find x from: (x+ ln(1/2x))=0

15. Jul 3, 2013

Tenshou

Is this the original question?

16. Jul 3, 2013

CallumC

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($\frac{1}{2x}$)

Find the value of x that gives the maximum speed.

17. Jul 3, 2013

Tenshou

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
18. Jul 3, 2013

micromass

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.

19. Jul 3, 2013

D H

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