I have a following function:
$f(x)=sin^2(a\cdot x)+sin^2(a\cdot (b-x))$
where $0\leq x \leq b$, $0<a\cdot b<\pi/2$, and $3\pi/2<a\cdot b<2\pi$. By exhaustive search I have noticed that function is strictly convex on the given range and that it has one extrema at $b/2$ which is a minimum while maximum is correspondingly attained at borders of the function ($0$ and b). My problem is that the function is discrete so I can not use the first and the second derivative for finding the minimum and maximum. Does anyone has any idea how to prove that my maximum is achieved at borders?
