Derivative at Point of Inflection

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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