Real function/Derivative/Monotonicity/Fact

Let ${}I\subseteq \mathbb {R}$ be an open interval, and let

$f\colon I\longrightarrow \mathbb {R}$

be a differentiable function. Then the following statements hold.

1. The function ${}f$ is increasing (decreasing) on ${}I$, if and only if ${}f'(x)\geq 0$ (${}f'(x)\leq 0$) holds for all ${}x\in I$.
2. If ${}f'(x)\geq 0$ holds for all ${}x\in I$, and ${}f'$ has only finitely many zeroes, then ${}f$ is strictly increasing.
3. If ${}f'(x)\leq 0$ holds for all ${}x\in I$, and ${}f'$ has only finitely many zeroes, then ${}f$ is strictly decreasing.

Proof, Write another proof