the graph of the first derivative f of a function f is shown. (assume the function is defined only for 0 ≤ x…

Explanation:

Step1: Recall increasing - decreasing property

A function $y = f(x)$ is increasing when $f'(x)>0$. Looking at the graph of $y = f'(x)$, we see that $f'(x)>0$ on the intervals $(1,4)$ and $(6,9)$.

Step2: Recall local - maximum property

A function $y = f(x)$ has a local maximum at $x = c$ when $f'(x)$ changes sign from positive to negative at $x = c$. From the graph, $f'(x)$ changes from positive to negative at $x = 4$.

Step3: Recall local - minimum property

A function $y = f(x)$ has a local minimum at $x = c$ when $f'(x)$ changes sign from negative to positive at $x = c$. From the graph, $f'(x)$ changes from negative to positive at $x = 1$ and $x = 6$.

Step4: Recall concavity property

A function $y = f(x)$ is concave upward when $f''(x)>0$, which means $f'(x)$ is increasing. $f'(x)$ is increasing on the intervals $(0,2)$ and $(5,7)$. A function $y = f(x)$ is concave downward when $f''(x)<0$, which means $f'(x)$ is decreasing. $f'(x)$ is decreasing on the intervals $(2,5)$ and $(7,9)$.

Step5: Recall inflection - point property

The inflection points of $y = f(x)$ occur where $f''(x)=0$, which means $f'(x)$ has a local maximum or minimum. From the graph, $f'(x)$ has local extrema at $x = 2$, $x = 5$ and $x = 7$.

Answer:

(a) $(1,4)\cup(6,9)$ (b) $x = 4$ (c) $x = 1,6$ (d) $(0,2)\cup(5,7)$ (e) $(2,5)\cup(7,9)$ (f) $x = 2,5,7$