If f is continuous on (a,b) and f′(x) exists and satisfies f′(x)>0 except at one point x0∈(a,b), then f is increasing in (a,b).

Question

If $f$ is continuous on $(a,b)$ and $f'(x)$ exists and satisfies $f'(x) >0$ except at one point $x_0 \in (a,b),$ then show that $f$ is increasing function in $(a,b).$

Answer 1

Hint

Let $x_1<x_2\in (a,x_0).$ There exists $c\in (x_1,x_2)$ such that $$f(x_2)-f(x_1)=f'(c)(x_2-x_1)>0.$$

Let $x_1\in (a,x_0).$ There exists $c\in (x_1,x_0)$ such that $$f(x_0)-f(x_1)=f'(c)(x_0-x_1)>0.$$