To determine the intervals where $f(x)$ is concave down, we analyze the curvature of the function $f(x)$ in the provided graph.

Key Concept

A function is concave down when its second derivative $f''(x) < 0$. On a graph, this means the slope of the tangent line (first derivative $f'(x)$) is decreasing.

Steps to Analyze Concavity:

1. Look for intervals where the slope of the tangent lines decreases (i.e., the graph curves downward).
2. Identify the turning points (where the graph changes from increasing to decreasing concavity) as key markers.

Observing the Graph:

From the graph:

1. Concave Down Intervals occur where the curve bends downwards, forming a "hill" shape.
2. Specifically:
   • The graph is concave down on the interval $(-7, -3)$ where the curve slopes upward but with decreasing slope.
   • The graph is concave down again on the interval $(2, 6)$ where the curve bends downward, forming a peak.

Final Answer:

The function $f(x)$ is concave down on the following intervals: $(-7, -3) \quad \text{and} \quad (2, 6).$

Let me know if you need further explanations or clarifications!

Related Questions:

1. How can you identify concavity using the second derivative $f''(x)$?
2. What happens at the points where $f''(x) = 0$?
3. How do you determine intervals of concavity from a table of slopes $f'(x)$?
4. How can you use inflection points to identify concavity changes?
5. How do the tangent line slopes relate to the shape of $f(x)$?

Tip:

To analyze concavity on a graph, focus on where the function's slope decreases (concave down) or increases (concave up).