An inflection point is a point between change in concavity. Concave-up corresponds to a positive second derivitive and concave-down corresponds to a negative second derivative. So when the function changes from concave-up to concave-down (or cancave- down to concave-up), the second derivative has to be equal to zero at that point. In order for the second derivative to be an inflection point it has to be equal to zero.
(Note: Derivative can either be referred to as the slope of a curve or as a rate of change.)
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