Question 27
Let $a, b \in \mathbb{R}$, $a \lt b$, and let $f : [a, b] \to [0, \infty)$ be a continuous nondecreasing function. Prove that
$$
\lim_{n \to \infty} \int_a^b \sqrt[n]{f^n(x) + f^n(a + b - x)} \, dx = 2 \int_{\frac{a+b}{2}}^b f(x) \, dx.
$$
$$
\lim_{n \to \infty} \int_a^b \sqrt[n]{f^n(x) + f^n(a + b - x)} \, dx = 2 \int_{\frac{a+b}{2}}^b f(x) \, dx.
$$