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Let's say I try to use the 2nd derivative test to find a local extrema at c.

To use the 2nd derivative test, must the 2nd derivative be continuous everywhere, or just a small interval near c?

Eg. Take the function $|x^2-1|$. The 2nd derivative is undefined at $x=1$ and $x=-1$, but there is a local maxima at x=0. Can I use the 2nd derivative test to show this?

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As the name already indicates, being a local extremum is a local property. Indeed, by definition, $f$ has a local maximum at $c$ if there is an $\varepsilon > 0$ such that $f(c) \geq f(x)$ for all $x \in (c-\varepsilon,c+\varepsilon)$. It thus suffices to consider the function on $(c-\varepsilon,c+\varepsilon)$. In your example you would indeed obtain a local maximum at $0$ by the second derivative criterion. But obviously the criterion says nothing about the extrema in $\pm 1$.

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