Math and science::Analysis::Tao::10: Differentiation of functions
Derivative of monotone functions
Monotone increasing implies a non-negative derivative
Let \( X \) be a subset of \( \mathbb{R} \), and let \( x_0 \in X \) be a limit point of \( X \). Let \( f: X \to \mathbb{R} \) be a function. If \( f \) is monotone increasing and is differentiable at \( x_0 \), then \( f'(x_0) \ge 0 \).
If instead, \( f \) is monotone decreasing, then \( f'(x_0) \le 0 \).
Strictly monotone doesn't imply strict derivative
Strictly monotone increasing/decreasing does not imply a strictly positive/negative derivative.
Strict derivative implies strictly monotone
We do have the result that if a function has a strictly positive derivative, then it must be strictly monotone increasing.
What about non-negative derivative implying monotone increasing? Tao doesn't quite explain if this is true or not. If it is true, you might have to include differentiability of the function on the whole domain.