Stuck with Optimisation question, help?

[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.

Please help and explain how to do it thanks :)

 

[SteamKing]

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

 

[Number Nine]

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

 

[CallumC]

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

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

 

[CallumC]

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

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

 

[SteamKing]

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.

 

[Tenshou]

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

 

[CallumC]

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.

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

 

[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

 

[CallumC]

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

Confused can someone write out the correct solution?

 

[Tenshou]

Is that the original problem?

 

[SteamKing]

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.

 

[CallumC]

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.

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?

 

[CallumC]

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

 

[Tenshou]

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

Is this the original question?

 

[CallumC]

Is this the original question?

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.

 

[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...

 

[micromass]

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...

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.

 

[D H]

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