Integrals inequality

This follows from the AM–GM inequality. Both integrals are positive, so $$A+B=4\implies\frac{A+B}2=2\implies \sqrt{AB}\le2\implies AB\le4$$

While Parcly's answer is quick, it can be done without AM-GM as well if you do not happen to know/remember AM-GM. After all, if given $A+B=4$, then the expression $AB$ can be written as $A(4-A)$ which is $-A^2+4A$. The outputs are values according to a parabola that opens down, having vertex $(2,4)$, hence $AB\le4$. Just a thought...