Theorem 4.10 goes like this . Let f :[a, b]→R be continuous and differentiable on (a, b). If f '(x)≥0 for all x∈(a, b), then f is monotonically increasing. Likewise, f is strictly increasing, monotonically decreasing, and strictly decreasing if ‘≥’ is replaced by ‘>’, ‘≤’, and ‘<’, resp.

i tried fucntions like x^2 , x^3, sin(x), cos(x), but none seemed to work ( FAILED TO PROVE THAT F'(X)>0)

Any help would be greatly appreciated