I dont know how you conducted the first devaritive test, but observe that g'(x)>0 for 0<x<(pi/4)
and g'(x)<0 for pi/4<x<5pi/4. Cos(x) decides the sign because exp(2/3sin(x)) is always > 0.
That means that on the interval 0<x<(pi/4) there is a local maxima, and on the interval pi/4<x<5pi/4, there is a local minima.
Looks like you just turned that one around.