The question asks to find the open intervals where the derivative $f'(x)$ is negative, based on the graph of $f(x)$ shown.

To find where $f'(x)$ is negative, we need to look at the intervals where the graph of $f(x)$ is decreasing. This is because the derivative $f'(x)$ represents the slope of the function, and where the slope is negative, the function is decreasing.

From the graph, we can observe the following:

• $f(x)$ is decreasing between approximately $x = -3$ and $x = 1$, where the curve slopes downward.

So the open interval where $f'(x)$ is negative is: $(-3, 1)$

This is the interval where $f(x)$ is decreasing.

Would you like more details or clarifications on this? Here are five follow-up questions related to this problem:

1. What is the significance of the derivative being positive or negative for a function?
2. How can we determine critical points on a graph like this?
3. What does it mean if the derivative is zero at a point?
4. How would you find where the function is increasing using the graph of $f(x)$?
5. How do you interpret inflection points in terms of the second derivative?

Tip: Remember that the sign of the derivative tells you whether a function is increasing or decreasing. If the derivative is negative, the function is going down; if it's positive, the function is going up.