Theorem 6.5.12: Local Extrema and Monotonicity
If f'(x) > 0 on (a, b) then f is increasing on (a, b).
If f'(x) < 0 on (a, b) then f is decreasing on (a, b).
This proof is left as an exercise. For a hint on the first part you might want to look at the proof of Rolle's theorem. If you have proved the first part, it should be more or less clear how the remaining parts can be proved. Recall the definitions of an increasing or decreasing function and compare it with the difference quotient involved in the derivative of f.