Derivative of Monotone Function

Theorem
Let $$f$$ be a real function which is continuous on the closed interval $$\left[{a \,. \, . \, b}\right]$$ and differentiable on the open interval $$\left({a \, . \, . \, b}\right)$$.


 * If $$\forall x \in \left({a \, . \, . \, b}\right): f^{\prime} \left({x}\right) \ge 0$$, then $$f$$ is increasing on $$\left[{a \, . \, . \, b}\right]$$.
 * If $$\forall x \in \left({a \, . \, . \, b}\right): f^{\prime} \left({x}\right) > 0$$, then $$f$$ is strictly increasing on $$\left[{a \, . \, . \, b}\right]$$.


 * If $$\forall x \in \left({a \, . \, . \, b}\right): f^{\prime} \left({x}\right) \le 0$$, then $$f$$ is decreasing on $$\left[{a \, . \, . \, b}\right]$$.
 * If $$\forall x \in \left({a \, . \, . \, b}\right): f^{\prime} \left({x}\right) < 0$$, then $$f$$ is strictly decreasing on $$\left[{a \, . \, . \, b}\right]$$.

Proof
Let $$c, d \in \left[{a \,. \, . \, b}\right]: c < d$$.

Then $$f$$ satisfies the conditions of the Mean Value Theorem on $$\left[{c \,. \, . \, d}\right]$$.

Hence $$\exists \xi \in \left({c \, . \, . \, d}\right): f^{\prime} \left({\xi}\right) = \frac {f \left({d}\right) - f \left({c}\right)} {d - c}$$.

If $$\forall x \in \left({a \, . \, . \, b}\right): f^{\prime} \left({x}\right) \ge 0$$, then $$f^{\prime} \left({\xi}\right) \ge 0$$ and hence $$f \left({d}\right) \ge f \left({c}\right)$$.

Thus $$f$$ is increasing on $$\left[{a \,. \, . \, b}\right]$$

If $$\forall x \in \left({a \, . \, . \, b}\right): f^{\prime} \left({x}\right) > 0$$, then $$f^{\prime} \left({\xi}\right) > 0$$ and hence $$f \left({d}\right) > f \left({c}\right)$$.

Thus $$f$$ is strictly increasing on $$\left[{a \,. \, . \, b}\right]$$.

The other cases follow similarly.