If $f''(x)<0$ for all $x \in(a, b)$, then $f'(x)=0$

at most once in (a, b)

If possible, let $x_1, x_2$ be two distinct points in $(a, b)$ such that $f^{\prime}\left(x_1\right)=f^{\prime}\left(x_2\right)=0$. Then, by Rolle's theorem there exists a point $c \in(a, b)$ such that $f^{\prime \prime}(c)=0$. This contradicts the given condition that $f^{\prime \prime}(x)<0$ for all $x \in(a, b)$.

Hence, our supposition is wrong. Consequently, there can be at most one point in $(a, b)$ at which $f^{\prime}(x)$ is zero.