Answer
See explanations.
Work Step by Step
Step 1. The given conditions for the function satisfy Rolle's Theorem which states that if there are two zeros in $f$, there should be at least one zero in $f'$ in between.
Step 2. Since it is given that $f'(x)\ne0$ in $(a,b)$, we do not have two zeros in $f(x)$ (one or no zeros).
Step 3. Because $f(a)$ and $f(b)$ have opposite signs, $f(x)$ must cross the x-axis ($f(x)=0$), but only once (one zero).
