# Derivative at Point of Inflection

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## Contents

## Theorem

Let $f$ be a real function which is twice differentiable on the open interval $\left({a \,.\,.\, b}\right)$.

Let $f$ have a point of inflection at $\xi \in \left({a \,.\,.\, b}\right)$.

Then:

- $f'' \left({\xi}\right) = 0$

where $f'' \left({\xi}\right)$ denotes the second derivative of $f$ at $\xi$.

## Proof

By definition of point of inflection, $f'$ has either a local maximum or a local minimum at $\xi$.

From Derivative at Maximum or Minimum, it follows that the derivative of $f'$ at $\xi$ is zero, that is:

- $f'' \left({\xi}\right) = 0$

$\blacksquare$

## Historical Note

The Derivative at Point of Inflection was given by Gottfried Wilhelm von Leibniz in his $1684$ article *Nova Methodus pro Maximis et Minimis*, published in *Acta Eruditorum*.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.19$: Leibniz ($\text {1646}$ – $\text {1716}$)