Notation of $\max$

Question

What is the meaning of:

\begin{equation} \max_{x_0\le x\le x_2}f^{\prime \prime}(x) \end{equation}

Is it the max of the second derivative at any $x$ between $x_0$ and $x_2$

Answer 1

Yes and if you know the second derivative of $f$ is continuous, then using max actually makes sense because continuous functions on closed intervals actually achieve their maximum by the extreme value theorem. If the interval weren't closed or the second derivative wasn't continuous, using "max" instead of "sup" would be risky notation.