Prove that $a^2x + b^2y \geq 2abc$, where $c \leq x, c \leq y$.
I've proved that $a^2 + b^2 \geq 2ab$, but I don't know how to deal with the coefficient terms. The question seems logical, given that $c$ is less than both $x$ and $y$, but I want to prove this more rigorously.