Question 51
The function $f: [a, b] \to \mathbb{R}$ is continuous and $f(y) \le \max\{f(x), f(z)\}$ for every $x, y, z \in [a, b]$ so that $x \lt y \lt z$. Denote $m = \min_{x \in [a, b]} f(x)$ and define the set $A = \{x \in [a, b] \mid f(x) = m\}$.
a) Prove that $A$ is a closed interval, possibly degenerated.
b) Prove that, if $a, b \notin A$, then there exists $c \in (a, b)$ such that on one of the intervals $[a, c]$ and $[c, b]$ the function is increasing and on the other it is decreasing.
a) Prove that $A$ is a closed interval, possibly degenerated.
b) Prove that, if $a, b \notin A$, then there exists $c \in (a, b)$ such that on one of the intervals $[a, c]$ and $[c, b]$ the function is increasing and on the other it is decreasing.