Question 22
Let $f$ and $g$ be continuous real-valued functions on the closed unit interval $[0, 1]$ such that $f(x)g(x) \ge 4x^2$ for all $x$ in $[0, 1]$. Show that (at least) one of the integrals
$$
\int_{0}^{1} f(x) \, dx, \quad \int_{0}^{1} g(x) \, dx
$$
has an absolute value greater than or equal to $1$.
$$
\int_{0}^{1} f(x) \, dx, \quad \int_{0}^{1} g(x) \, dx
$$
has an absolute value greater than or equal to $1$.