Morgan D.

asked • 04/01/16

Finding the Min & Max of a function

Question: "for the following functions, determine the locations of relative extrema, either minimum or maximum, and find their values"
 
for k(x)=(ln3x)/(2x^2)
 
The process I use is to first find the derivative --> find the critical #'s --> find 2nd derivative & evaluate at critical numbers --> test critical numbers in original equation to get the value of the min and/or max
 
However, I can't seem to get this problem right no matter how many times I've tried...I think it's because I'm not finding the correct derivatives.
 
When I finally looked up the answer I got max @ e/sqrt3 and no minima found...can someone please tell me if this is correct or if it's not correct what the correct answer is? Or if it's correct what the steps to getting to that answer would be?
 
Thank you SO much!

1 Expert Answer

By:

Victoria V. answered • 04/01/16

Tutor
5.0 (399)

20+ years teaching Calculus

Mark M.

The Chain Rule is used to determine the derivative of a composite function h(x) = f ( g (x )).
The function given is merely a division/product.
k(x) = (0.5)(ln3x)(x-2)
The Product Rule: 
h(x) = f(x)g(x)
h'(x) = [f(x)g(x)]'
h'(x) = f'(x)g(x) + f(x)g'(x)
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04/01/16

Philip P.

ln(3x) is a composite function and requires the chain rule.
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04/01/16

Victoria V.

Thank you, Phillip.
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04/01/16

Victoria V.

To help my students remember to do the chain rule - I have them do it on EVERY derivative, and I will explain this:
 
Let's say that f(x) = x2 so f'(x) = 2x.  Easy enough.  I know, you do not NEED the chain rule.  But what if you looked at the problem like this:
 
If you treat this as a general power rule, f(x) = (x)2,  then when you take the derivative you get f'(x) = 2 (x)1(1) = 2x.  Same thing, but the students see that the chain rule ALWAYS applies, it is just not necessary with a "plain old 'x'". 
 
I have had students ask why the chain rule applies to some problems but not others, and so I do the chain rule EVERY time so that they don't forget to do it when it is absolutely necessary.
 
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04/01/16

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