A= sin^{2}x + cos^{4}x
It may be worthwhile in general to confirm via the second derivative test as such:
Re-writing dA/dx as -sin(2x)cos(2x)= -sin(4x)/2, we can easily compute the second derivative: -2cos(4x)
At x=0, the second derivative is <0, and thus this is indeed a maximum-generating point
At x= pi/4, the second derivative is >0 and thus this is indeed a minimum-generating point
(no boundary value conditions stated)