Question

use the graph of f(x) to sketch f(x).

Explanation:

Step1: Identify increasing - decreasing intervals

Where $f(x)$ is increasing, $f'(x)>0$. Where $f(x)$ is decreasing, $f'(x)<0$. $f(x)$ is increasing on $(-\infty, 0)$ and $(3,5)$ and decreasing on $(0,3)$.

Step2: Identify critical points

Critical points of $f(x)$ occur where $f'(x) = 0$. The critical points of $f(x)$ are at $x = 0$ and $x=3$ since the slope of $f(x)$ is 0 at these points. So $f'(x)$ will cross the x - axis at $x = 0$ and $x = 3$.

Step3: Analyze concavity

The concavity of $f(x)$ affects the slope of $f'(x)$. When $f(x)$ is concave up, $f'(x)$ is increasing and when $f(x)$ is concave down, $f'(x)$ is decreasing. $f(x)$ is concave down on $(-\infty,1.5)$ and concave up on $(1.5,5)$. So $f'(x)$ has a local maximum at $x = 1.5$.

Answer:

Sketch $f'(x)$ such that it is positive on $(-\infty,0)\cup(3,5)$, negative on $(0,3)$, crosses the x - axis at $x = 0$ and $x = 3$, and has a local maximum at $x=1.5$.