the graph of $f(x)$ is given below. on what interval(s) is the value of the derivative $f(x)$ negative? give…

Explanation:

Step1: Identify where the function is decreasing.

The derivative $f'(x)$ is negative where the function $f(x)$ is decreasing. Observe the graph to find these intervals. The function $f(x)$ decreases from the far left until it reaches a local minimum. Then it increases until it reaches a local maximum, after which it decreases again.

Step2: Determine the x-values of local minimum and maximum.

From the graph, the function $f(x)$ has a local minimum at $x = -4$. The function $f(x)$ has a local maximum at $x = 0$.

Step3: Write the intervals where the function is decreasing.

The function $f(x)$ is decreasing on the interval $(-\infty, -4)$. The function $f(x)$ is also decreasing on the interval $(0, \infty)$. Therefore, the derivative $f'(x)$ is negative on these intervals.

Answer:

$(-\infty, -4) \cup (0, \infty)$