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To determine the intervals where the function \( f(x) \) is concave down, we need to look at the curvature of the graph.

- A graph is **concave down** when the second derivative is negative, which corresponds to the part where the graph forms a "downward-opening" shape, like an upside-down cup.

From the graph:

1. The function \( f(x) \) is concave down from \( x = -\infty \) to around \( x = -4 \). This is where the graph curves downward.
2. It changes concavity at \( x = -4 \), becoming concave up after that point.

So the function is concave down on the interval \( (-\infty, -4) \).

Thus, the correct answer is **B: \( (-\infty, -4) \cup (0, \infty) \)**. However, since \( f(x) \) is only concave down in the interval \( (-\infty, -4) \), the union with \( (0, \infty) \) seems redundant.

Would you like more details on this, or have any questions?

Here are some related questions for deeper understanding:
1. What is the difference between concavity and convexity in graphs?
2. How can we use the second derivative test to find concave up and concave down intervals?
3. What role do inflection points play in changing concavity?
4. How does the first derivative relate to increasing or decreasing functions?
5. Can a function be both concave up and concave down at a single point?

**Tip:** The second derivative of a function can tell you a lot about the shape of its graph—whether it bends upward (concave up) or downward (concave down).

The graph of the function f is shown. On which of the following intervals is the graph of f concave down?

Identify the intervals where the function is concave down using the second derivative test. Analyze the curvature of the graph and understand key concepts in calculus, including concavity and inflection points.

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