Proofs of Derivative Applications Facts — Question 4

Question 4 - Solution

To show that $f$ is increasing on $(a,b)$, we must prove that for any $x_1,x_2\in(a,b)$ with $x_1<x_2$, we have $$f(x_1)<f(x_2).$$

Let $x_1,x_2\in(a,b)$ with $x_1<x_2$. Since $f$ is differentiable on $(a,b)$, it is continuous on $[x_1,x_2]$ and differentiable on $(x_1,x_2)$.

By the Mean Value Theorem, there exists a number $c\in(x_1,x_2)$ such that $$f'(c)=\frac{f(x_2)-f(x_1)}{x_2-x_1}.$$

By assumption, $f'(c)>0$, and since $x_2-x_1>0$, it follows that $$\frac{f(x_2)-f(x_1)}{x_2-x_1}>0.$$

Multiply both sides by $x_2-x_1$: $$f(x_2)-f(x_1)>0.$$

Thus, $$f(x_2)>f(x_1).$$

Since this holds for all $x_1<x_2$ in $(a,b)$, the function $f$ is increasing on $(a,b)$.

$$\boxed{f\text{ is increasing on }(a,b)}$$