Answer
No.
Counterexample: $\quad f(x)=x^{2}$ on $[-3,1 ]$
(sample answer)
Work Step by Step
For a counterexample, select a function so that $f^{\prime}(0)=0$, such as $f(x)=x^{2}$,
and then select an interval $[a,b]$ so that the function values of endpoints are not symmetric.
For example $[-3,1 ],\quad f(-3)=9,\quad f(1)=1$
So, with$\quad f(x)=x^{2}$ on $[-3,1 ]$
we have $\quad f^{\prime}(x)=2x$ and $f^{\prime}(0)=0$
$ 0\in (-3,1)$ but $f(-3)\neq f(1)$ .
