Sufficient Condition for Stationary Point to be Local Maximum

Theorem

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

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

Let the second derivative of $f$ at $\xi$ be (strictly) negative.

Then $\xi$ is a local maximum.

Proof

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Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Maximum, Minimum and Point of Inflection
• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.4$ Limits, Maxima and Minima: $3.4.2 \ (1)$