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To solve this problem, we need to analyze the given function \( f(x) = \frac{-4}{x^2 + 4} \) and its derivatives:

1. **Function and Derivatives Analysis:**
   - The function \( f(x) \) is defined as \( f(x) = \frac{-4}{x^2 + 4} \).
   - Its first derivative, \( f'(x) = \frac{8x}{(x^2 + 4)^2} \), gives us information about increasing or decreasing behavior.
   - The second derivative, \( f''(x) = \frac{-24x^2 + 32}{(x^2 + 4)^3} \), reveals concavity.

2. **Task (a): Identifying the Correct Graph:**
   - The graph showing \( f(x) \) in blue (thick line) and \( f''(x) \) in red (thin line) needs to match the expected shape and behavior derived from the concavity.
   - \( f(x) \) should approach zero as \( x \to \pm \infty \) because of the denominator in \( x^2 + 4 \), which causes the function to decay.
   - Since \( f(x) \) has negative values only, we expect it to have a minimum but no maximum, and it should remain entirely below the x-axis.
   - After inspecting the graphs, **Graph D** matches this description correctly.

3. **Task (b): Concavity Intervals:**
   - The sign of \( f''(x) \) determines the intervals of concavity:
     - **Concave up** where \( f''(x) > 0 \)
     - **Concave down** where \( f''(x) < 0 \)
   - Setting \( f''(x) = 0 \), we solve for intervals of concavity:
     - \( f''(x) = 0 \Rightarrow -24x^2 + 32 = 0 \Rightarrow x = \pm \sqrt{\frac{4}{3}} \approx \pm 1.5 \)
   - Therefore:
     - \( f \) is concave up on \( (-1.5, 1.5) \).
     - \( f \) is concave down on \( (-\infty, -1.5) \cup (1.5, \infty) \).

The solutions are:
- **(a)** Graph **D**.
- **(b)** Concave up on \( (-1.5, 1.5) \) and concave down on \( (-\infty, -1.5) \cup (1.5, \infty) \).

Would you like further explanation on any part of this solution?

Here are five related questions:
1. What does the first derivative \( f'(x) \) tell us about the function's behavior?
2. How does the denominator in \( f(x) \) influence its end behavior?
3. Why is concavity important in understanding the shape of a graph?
4. How can inflection points be identified from \( f''(x) \)?
5. How does the symmetry of \( f(x) \) affect the analysis of intervals?

**Tip:** When analyzing graphs, identifying where derivatives are zero can reveal critical points for both maxima, minima, and inflection points.

Consider the function f(x) = -4 / (x^2 + 4) and its derivatives. Identify the graph displaying f in blue (thick) and f'' in red (thin), and indicate the x-intervals where f is concave up and concave down.

Analyze the function f(x) = -4 / (x^2 + 4) and its derivatives to determine concavity intervals. Identify the correct graph for f in blue and f'' in red, and specify intervals where f is concave up and down. Suitable for advanced calculus students in Grades 11-12.

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