the graph of f is shown in the figure. (a) for which values of x is f(x) zero? (enter your answers as a…

Explanation:

Step1: Recall derivative - slope relationship

The derivative $f^{\prime}(x)$ is zero where the graph of $y = f(x)$ has a horizontal tangent.

Step2: Identify horizontal - tangent points

From the graph, the function $y = f(x)$ has horizontal tangents at $x=- 2,2$. So for $f^{\prime}(x)=0$, $x=-2,2$.

Step3: Determine where $f^{\prime}(x)>0$

The function $y = f(x)$ is increasing when $f^{\prime}(x)>0$. From the graph, $f(x)$ is increasing on the intervals $(-\infty,-2)$ and $(2,\infty)$.

Step4: Determine where $f^{\prime}(x)<0$

The function $y = f(x)$ is decreasing when $f^{\prime}(x)<0$. From the graph, $f(x)$ is decreasing on the interval $(-2,2)$.

Step5: Recall second - derivative meaning

The second - derivative $f^{\prime\prime}(x)$ is zero where the graph of $y = f(x)$ has an inflection point (where the concavity changes).

Step6: Identify inflection points

From the graph, the inflection points occur at $x = 0$. So for $f^{\prime\prime}(x)=0$, $x = 0$.

Answer:

(a) $x=-2,2$; $(-\infty,-2)\cup(2,\infty)$; $(-2,2)$ (b) $x = 0$