Chapter 3 - Applications of Differentiation - 3.3 How Derivatives Affect the Shape of a Graph - 3.3 Exercises - Page 231: 71

See proof

By hypothesis $g = f'$ is differentiable on an open interval containing $c$. Since $(c,f(c))$ is a point of inflection, the concavity changes at $x = c$, so $f''(x)$ changes signs at $x = c$. Hence, by the First Derivative Test, $f'$ has a local extremum at $x = c$. Thus, by Fermat’s Theorem $f''(c) = 0$.