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Using the 2nd derivative test, we know that if the value from evaluating critical points at the 2nd derivative is greater than 0, then it is a local min. If the value is less than 0, then it is a local max.

However, if the value is = 0, then it is "inconclusive".

What does this mean? Is it considered an inflection point/saddle point?

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Careful: $f''(c) > 0$ implies local min, not max.

It may not be an inflection point; this all depends on the particular function. It is indeed inconclusive. Consider $f(x) = x^4$. Clearly $x=0$ is critical, but $f''(0)=0$. But still, $x=0$ clearly gives a local min and is not an inflection point. The function $f(x)=x^3$ shows that the critical point may be an inflection point.

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  • $\begingroup$ ahh, edited**** $\endgroup$ – mathguy Nov 17 '17 at 3:20

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