Proofs of Derivative Applications Facts — Question 5

Question 5 - Solution

To prove that $f$ is decreasing on $(a,b)$, we must show that for any $x_1,x_2\in(a,b)$ with $x_1<x_2$, the inequality $$f(x_1)>f(x_2)$$ holds.

Let $x_1,x_2\in(a,b)$ with $x_1<x_2$. Because $f$ 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 decreasing on $(a,b)$.

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