Derivative at Point of Inflection

Theorem
Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$.

Let $f$ have a point of inflection at $\xi \in \openint a b$.

Then:
 * $\map {f''} \xi = 0$

where $\map {f''} \xi$ denotes the second derivative of $f$ at $\xi$.

Proof
By definition of point of inflection, $f'$ has either a local maximum or a local minimum at $\xi$.

From Derivative at Maximum or Minimum, it follows that the derivative of $f'$ at $\xi$ is zero, that is:
 * $\map {f''} \xi = 0$