Horizontal Point of Inflection is Stationary Point

Theorem

Let $f$ be a real function which is differentiable on an interval $\Bbb I \subseteq \R$.

Let $\xi \in \Bbb I$ be such that $\xi$ has a point of inflection at $\xi$ such that the tangent to $f$ at $\xi$ is parallel to the $x$-axis.

Then $\xi$ is a stationary point.

Proof

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Sources

• 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