In Baby Rudin, Theorem 5.11 says, Suppose $f$ is differentiable in $(a,b)$. If $f'(x) \geq 0$ for all $x \in (a,b)$, then $f$ is monotonically increasing, but this is an if and only if, right?
If we analyze the behavior of a monotonically increasing differentiable function, then we realize that $\frac{f(y) - f(x)}{y-x}$ is always nonnegative. So, if $f'(x)$ exists, then $f'(x) \geq 0$, right?
Yes dude you are right. Here’s an extended problem: what’s the behavior of $f’(x)$ for strictly monotonically increasing function?