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I know that where the first derivative equals 0 or doesn't exist, we might have a local min or max. We can plug these critical numbers in to the second derivative and depending if we get positive or negative answer, that tells us if the point is in fact a local min or max

Well, where the second derivative equals 0 or doesn't exist we might have a point of inflection. Can we plug these critical numbers in to the third derivative and depending if we get positive or negative, verify such a result?

Thanks