I can't figure this out at all, I've got the derivative

$\displaystyle g'(x)=\frac{1}{2}(4cos^{2}x+4cosx-3)e^{\frac{1}{2}x+cosx}$

and the domain of the function $\displaystyle g$ is $\displaystyle [0,2\pi]$

I've factorised to get

$\displaystyle (2cosx-1)(2cosx+3)=0$

But when I put the second bracket as zero to find $\displaystyle x$ for that bracket I get

$\displaystyle x=arccos(\frac{-3}{2})$

which doesn't get me anywhere on my calculator.

Have I made a mistake somewhere?

Many thanks.