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?

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 $x \to \pm \infty$.