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To determine where the graph is concave down, we need to look for intervals where the graph of the function \( f(x) \) curves downward, meaning its second derivative \( f''(x) < 0 \). In simpler terms, concavity down happens when the graph looks like a downward-facing bowl or hill.

From the graph:

- The curve is concave down when it is sloping downward and becoming less steep. This occurs from \( x = -\infty \) up to about \( x = -2 \).
- After \( x = -2 \), the graph switches to concave up, forming an upward curve.

Thus, the graph is concave down on the interval \( (-\infty, -2) \).

**Correct answer: Option A** \( (-\infty, -2) \).

Would you like further details on concavity or how to determine concave intervals?

Here are 5 related questions you might explore:
1. What is the difference between concave up and concave down?
2. How do we use the second derivative to determine concavity?
3. Can you determine points of inflection from this graph?
4. What happens to the first derivative at concave down intervals?
5. How does concavity relate to acceleration in physical problems?

**Tip:** If you see the graph bending downward, it's likely concave down—use the curve's shape as a visual indicator!

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

Explore how to determine concave down intervals of the function f from its graph. This problem demonstrates the use of the second derivative test and concavity theorem to identify intervals where the graph curves downward, such as (-∞, -2). Ideal for students in Grades 11-12.

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