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.

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

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