# 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.