the graph of the derivative f of a function f is shown. (a) on what intervals is f increasing? (enter your…

Explanation:

Step1: Recall increasing - decreasing property

A function $y = f(x)$ is increasing when $f'(x)>0$ and decreasing when $f'(x)<0$.

Step2: Find intervals of increase

Looking at the graph of $y = f'(x)$, $f'(x)>0$ on the intervals $(0, 2)$ and $(4, 6)$. So the function $f(x)$ is increasing on the intervals $(0, 2)\cup(4, 6)$.

Step3: Find intervals of decrease

$f'(x)<0$ on the intervals $(-\infty,0)$ and $(2, 4)$. So the function $f(x)$ is decreasing on the intervals $(-\infty,0)\cup(2, 4)$.

Step4: Recall local - extrema property

Local maxima and minima occur where $f'(x)$ changes sign. $f'(x)$ changes from positive to negative at $x = 2$, so $f(x)$ has a local maximum at $x = 2$. $f'(x)$ changes from negative to positive at $x=0$ and $x = 4$, so $f(x)$ has local minima at $x = 0$ and $x = 4$.

Answer:

(a) Increasing intervals: $(0, 2)\cup(4, 6)$; Decreasing intervals: $(-\infty,0)\cup(2, 4)$ (b) $x=0,2,4$