I am trying to figure out where the maximum occurs for the following expression of 2 variables ($x$ and $y$):
$x(C-x)A^2+y(D-y)B^2+(x(D-y)+y(C-x))AB$
where $-1\leq A,B\leq 1$, $C\geq 2$ and $D\geq 2$ are all constants. Moreover the range of values allowed for $x$ and $y$ is $0\leq x\leq C$ and $0\leq y\leq D$.
I tried taking partial derivatives but the second order partial derivative test (https://en.wikipedia.org/wiki/Second_partial_derivative_test) is inconclusive so I was wondering if there may be another way to approach this.
Hint: write it as $\;(xA+yB)\big((C-x)A+(D-y)B\big)\,$ and use that $\,ab \le \left(\frac{a+b}{2}\right)^2\,$.