In a script I'm reading right now that is providing a proof for Cauchy's mean value theorem (extended mean value theorem), it says
"We have $g(x) \neq 0$ for all $x\in ]a,b[$ (otherwise, there would be a point $x_0\in]a,b[$ with $g^{\prime} = 0$ - which is a contradiction)"
The contradiction stems from the fact that we stated $g^{\prime} \neq 0$ before, I'm just really wondering about the string of thought. Does Rolle's Theorem really suggest that if a function goes through zero, it must have an extremum (which wouldn't make sense to me). Thanks for any hints on if I'm missing something.