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Let's work through this problem, which involves analyzing the function:

\[
f(x) = \frac{2x^2}{x^2 - 9}
\]

### Step-by-Step Solutions:

#### (a) First Derivative and Critical Points
1. **First Derivative**: Use the quotient rule for differentiation, which states that if \( f(x) = \frac{g(x)}{h(x)} \), then:
   \[
   f'(x) = \frac{g'(x) h(x) - g(x) h'(x)}{(h(x))^2}
   \]
   Here, \( g(x) = 2x^2 \) and \( h(x) = x^2 - 9 \).

2. **Critical Points**: Set \( f'(x) = 0 \) and solve for \( x \). The critical points occur where \( f'(x) = 0 \) or where \( f'(x) \) is undefined (i.e., where \( h(x) = 0 \)).

3. **Intervals of Increase/Decrease**: Determine where \( f'(x) > 0 \) (increasing) and \( f'(x) < 0 \) (decreasing).

4. **Local Extrema**: Use the sign changes of \( f'(x) \) to find local maxima and minima.

#### (b) Vertical Asymptotes and Limits
1. **Vertical Asymptotes**: Set the denominator equal to zero, \( x^2 - 9 = 0 \), to find vertical asymptotes at \( x = 3 \) and \( x = -3 \).

2. **Left- and Right-Hand Limits**:
   - Calculate the left-hand and right-hand limits of \( f(x) \) as \( x \) approaches each asymptote:
     \[
     \lim_{x \to -3^-} f(x), \quad \lim_{x \to -3^+} f(x), \quad \lim_{x \to 3^-} f(x), \quad \lim_{x \to 3^+} f(x)
     \]

3. **Limits at Infinity**: Compute \( \lim_{x \to \infty} f(x) \) and \( \lim_{x \to -\infty} f(x) \) to find any horizontal asymptotes.

#### (c) Second Derivative and Concavity
1. **Second Derivative**: Differentiate \( f'(x) \) to obtain \( f''(x) \).

2. **Concave Up/Down Intervals**: Identify where \( f''(x) > 0 \) (concave up) and \( f''(x) < 0 \) (concave down).

3. **Inflection Points**: Find where \( f''(x) = 0 \) or is undefined and where the concavity changes.

#### (d) Symmetry of the Function
Determine if \( f(x) \) is even or odd by testing \( f(-x) \) and comparing it to \( f(x) \).

#### (e) Sketching the Graph
Based on the information above, sketch the graph with:
- Asymptotes,
- Critical points,
- Intervals of increase/decrease,
- Concave up/down regions,
- Inflection points.

Would you like detailed solutions for each part, or have specific questions about any step?

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### Follow-Up Questions:
1. How do you determine the concavity of a function using the second derivative?
2. What role do vertical asymptotes play in the behavior of a function?
3. How can you determine if a function is even or odd?
4. Why are limits at infinity important when sketching graphs?
5. What is the significance of critical points in identifying local extrema?

**Tip**: For asymptotes, remember that vertical asymptotes arise from the zeros of the denominator, while horizontal asymptotes depend on the behavior of \( f(x) \) as \( x \to \pm \infty \).

Analyze the function f(x) = (2x^2) / (x^2 - 9) to determine its first and second derivatives, critical points, intervals of increase/decrease, concavity, asymptotes, and graph symmetry.

Learn how to analyze the function f(x) = (2x^2) / (x^2 - 9) using calculus. Find critical points, intervals of increase and decrease, concavity, asymptotes, and determine graph symmetry. Suitable for college-level Calculus I students.

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