# Question on local extrema problem

• Jun 29th 2010, 04:43 PM
ascendancy523
Question on local extrema problem
Hey everyone, I have this problem that I need some help on:

Identify all critical points and use first derivative test and second derivative test to decide which of the critical points give a local max. and local min.

$\displaystyle f(x)=\frac{1}{2}x+sinx, 0<x<2\pi$

So,

$\displaystyle f'(x)=\frac{1}{2}+cosx$

$\displaystyle f''(x)=-sinx$

So $\displaystyle f'(x)=0$ when x is $\displaystyle \frac{2\pi}{3} , \frac{4\pi}{3}$?

Then from there plug the critical points into f(x) to find the man and min points?
• Jun 29th 2010, 04:53 PM
HallsofIvy
Any max and min must occur at those points and since there are exactly one of them, it is a good bet that one gives the maximum value of f and the other the minimum. I emphasized "value of f" to indicate that, yes, we are talking about the maximum and minimum of f(x). You can probably just see from the values which is the maximum and which the minimum value but since the problem asks you to use the second derivative, if f'(x)= 0 and f"(x)> 0, that x gives a minimum value of f(x). If f'(x)= 0 and f"(x)< 0, that x gives a maximum value of f(x).