Necessary 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 $\xi$ be a local maximum.

Then there exists an open interval $\openint c d$ such that:

\(\ds \xi\)

\(\in\)

\(\ds \openint c d\)

\(\ds \forall x \in \openint c \xi: \, \)

\(\ds \map {f'} x\)

\(>\)

\(\ds 0\)

\(\ds \forall x \in \openint \xi d: \, \)

\(\ds \map {f'} x\)

\(<\)

\(\ds 0\)

Proof

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Also see

• Necessary Condition for Stationary Point to be Local Minimum

• Sufficient Condition for Stationary Point to be Local Maximum
• Sufficient Condition for Stationary Point to be Local Minimum

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Maximum, Minimum and Point of Inflection
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): turning point
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): turning point