Second Derivative at Point of Inflection

Theorem

Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$.

Let $f$ have a point of inflection at $\xi \in \openint a b$.

Then:

$\map {f} \xi = 0$

where $\map {f} \xi$ 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:

$\map {f} \xi = 0$

$\blacksquare$

Also see

• Point with Zero Second Derivative is not necessarily Point of Inflection

Historical Note

The result Second 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

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Maximum, Minimum and Point of Inflection
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.19$: Leibniz ($\text {1646}$ – $\text {1716}$)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): inflection
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): inflection