Putnam 2013 B4

For any continuous real-valued function $f$ defined on the interval $[0,1]$, let $$\mu(f) = \int_0^1 f(x) dx,\ \text{Var}(f) = \int_0^1 (f(x) - \mu(f))^2 dx,\ \ M(f) = \max_{0 \leq x \leq 1} |f(x)|.$$ Show that if $f$ and $g$ are continuous real-valued functions defined on the interval $[0,1]$, then $$\text{Var}(fg) \leq 2\text{Var}(f)M(g)^2 + 2\text{Var}(g)M(f)^2.$$