Question

answer the following true or false: if f(x) is a differentiable function that is positive for all x, then f(x) is increasing for all x. true false

Explanation:

Step1: Recall derivative concept

The sign of $f(x)$ does not determine the behavior of $f'(x)$.

Step2: Consider a counter - example

Let $f(x)=e^{-x}$, $f(x)>0$ for all $x\in R$. And $f'(x)=-e^{-x}$, $f''(x) = e^{-x}>0$. Now let $f(x)= 2$, a constant function. $f(x)>0$ for all $x$, but $f'(x)=0$ which is neither increasing nor decreasing. So just because $f(x)>0$ for all $x$, we cannot say $f'(x)$ is increasing for all $x$.

Answer:

False