Suppose that a function \( f:[0, 1] \to (0, \infty) \) satisfies that
\[ \int_0^1 f(x) dx = 1. \]
Prove the following inequality.
\[ \left( \int_0^1 |f(x)-1| dx \right)^2 \leq 2 \int_0^1 f(x) \log f(x) dx. \]