Show that $f'(x) \neq 0$ for all $x \in (a,b)$ implies $f(p) = 0$ for at most one $p \in [a,b]$

Question

Suppose $f \in C[a,b]$ and $f'(x)$ exists on $(a,b)$. Show that if $f'(x)$ not equal to $0$ for all $x$ in $(a,b)$, then there can exist at most one number $p$ in $[a,b]$ with $f(p) = 0$.

Is there any theoretical proof available to this Theorem?

Answer 1

Assume that the derivative function $f'(x)$ is continuous on an interval. Then on that interval it is always positive or always negative, being never zero. Consequently $f(x)$ will be either increasing or decreasing throughout that interval. And so $f(x)$ cannot take the same value twice (zero or whatever).