# finding critical points of a fuction

• Dec 9th 2012, 01:59 PM
kingsolomonsgrave
finding critical points of a fuction
Attachment 26159

I first find the derivative of the function using the quotient rule, to get $\frac{1-lnx}{x^2}$ and set equal to zero and solve to get x=e

is this correct, or are there are there more steps?
• Dec 9th 2012, 03:41 PM
AZach
Re: finding critical points of a fuction
That looks right. x = 0 is a critical number too though because it causes f(c) to no longer exist, but that's outside the interval. Did you test the end points? There's a theorem (I think extreme value theorem) that says there should be an absolute maximum and minumum in that interval which means there'll be another critical number in that interval.
• Dec 9th 2012, 04:03 PM
skeeter
Re: finding critical points of a fuction
Quote:

Originally Posted by AZach
That looks right. x = 0 is a critical number too though because it causes f(c) to no longer exist, but that's outside the interval. Did you test the end points? There's a theorem (I think extreme value theorem) that says there should be an absolute maximum and minumum in that interval which means there'll be another critical number in that interval.

not true ... x = c is a critical value of f(x) if f(c) is defined and f'(c) = 0 or f'(c) is undefined. Endpoint extrema are not necessarily critical values. Recommend you check out the link re: definition of a critical value.

Pauls Online Notes : Calculus I - Critical Points
• Dec 9th 2012, 04:16 PM
AZach
Re: finding critical points of a fuction
Since 0 is outside the interval does that mean the same as it not being defined?
• Dec 9th 2012, 04:19 PM
skeeter
Re: finding critical points of a fuction
Quote:

Originally Posted by AZach
Since 0 is outside the interval does that mean the same as it not being defined?

no ... $f(x) = \frac{\ln{x}}{x}$ , $x = 0$ is not even in the domain of the function.
• Dec 9th 2012, 04:36 PM
AZach
Re: finding critical points of a fuction
Quote:

Originally Posted by skeeter
no ... $f(x) = \frac{\ln{x}}{x}$ , $x = 0$ is not even in the domain of the function.

Oh!