Question 67
Let $f: [0, 1] \to \mathbb{R}$ be a continuous function such that $f((1-t)x + ty) \le \max\{f(x), f(y)\}$, for every $t, x, y \in [0, 1]$, and let $m = \inf_{0 \le x \le 1} f(x)$. Show that:
a) the set $\{x \mid 0 \le x \le 1, f(x) = m\}$ is a closed interval;
b) if $f(0) = 0$, $\int_{0}^{1} f(x) \, dx = 0$ and $g: [0, 1] \to \mathbb{R}$ is a non-decreasing function, then
$$
\int_{0}^{1} f(x)g(x) \, dx \ge 0.
$$
a) the set $\{x \mid 0 \le x \le 1, f(x) = m\}$ is a closed interval;
b) if $f(0) = 0$, $\int_{0}^{1} f(x) \, dx = 0$ and $g: [0, 1] \to \mathbb{R}$ is a non-decreasing function, then
$$
\int_{0}^{1} f(x)g(x) \, dx \ge 0.
$$