The Mean Value Theorem

The Increasing Function Theorem

Suppose that f is continuous on [a, b] and differentiable on (a, b),

and that the derivative of f is positive on (a, b).

Then f is increasing on [a, b].

a b

Intuitive visual sketch of proof:

What would the Mean Value Theorem say about this?

There would have to be at least one place between those two points where the derivative of f would be negative.

This is impossible if the derivative of f is positive on (a, b).