Necessary Condition for Stationary Point to be Local Minimum

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Theorem

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

Let $\xi$ be a local minimum.

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

\(\ds \xi\) \(\in\) \(\ds \openint c d\)
\(\ds \forall x \in \openint c \xi: \, \) \(\ds \map {f'} x\) \(\ge\) \(\ds 0\)
\(\ds \forall x \in \openint \xi d: \, \) \(\ds \map {f'} x\) \(\le\) \(\ds 0\)


Proof



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