The graph provided shows the function $f(x)$. The problem asks to approximate the intervals where $f'(x) < 0$, meaning we need to find where the slope of the graph is negative (i.e., where the graph is decreasing).

Analysis:

• $f'(x)$, the derivative of $f(x)$, represents the slope of the tangent line to the graph at any point.
• The function is decreasing where the slope is negative, so we look for portions of the graph that are sloping downwards.

Approximate intervals where $f'(x) < 0$:

• From the graph, it appears that $f(x)$ is decreasing between:
  • $x \approx -3.2$ to $x \approx -1$ (left of the origin).
  • $x \approx 0.4$ to $x \approx 1.8$ (right of the origin).

Thus, the approximate intervals where $f'(x) < 0$ are: $(-3.2, -1) \quad \text{and} \quad (0.4, 1.8)$

Would you like more detailed explanations or have further questions?

Follow-up Questions:

1. What is the significance of $f'(x) > 0$ on the graph?
2. How do we identify critical points of $f(x)$ using $f'(x)$?
3. What do the intervals where $f'(x) = 0$ represent?
4. How does the behavior of $f(x)$ change at local maxima and minima?
5. How can we estimate inflection points from a graph?

Tip:

When identifying where a function is increasing or decreasing, look for intervals where the graph moves upward or downward.