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?