To sketch a graph of the function $f(x)$, let's use the provided information step by step:

Key Information Breakdown:

1. Intervals for $f'(x)$ and $f''(x)$:

   • For $x \in (-\infty, -2)$:
     $f'(x) > 0$ (increasing), $f''(x) > 0$ (concave up).
   • For $x \in (-2, 0)$:
     $f'(x) > 0$ (increasing), $f''(x) < 0$ (concave down).
   • For $x \in (0, 2)$:
     $f'(x) < 0$ (decreasing), $f''(x) < 0$ (concave down).
   • For $x \in (2, \infty)$:
     $f'(x) < 0$ (decreasing), $f''(x) > 0$ (concave up).

2. Vertical Asymptotes:

   • There are vertical asymptotes at $x = -2$ and $x = 2$.

3. Critical Point at $x = 0$:

   • $f(0) = 0$, so the graph passes through $(0,0)$.
   • $f'(0) = 0$ indicates a horizontal tangent (possible maximum or minimum) at $x = 0$.
   • $f''(0) < 0$ suggests that $x = 0$ is a local maximum, because the graph is concave down at this point.

Steps for the Sketch:

1. Start with the vertical asymptotes at $x = -2$ and $x = 2$. The graph will approach these asymptotes but never touch or cross them.
2. On the interval $(-\infty, -2)$, the function is increasing and concave up.
3. Between $(-2, 0)$, the function is still increasing but concave down. As $x \to -2$, the function will approach the vertical asymptote from below.
4. At $x = 0$, the graph has a maximum since $f(0) = 0$, $f'(0) = 0$, and $f''(0) < 0$.
5. Between $(0, 2)$, the function decreases and is concave down. As $x \to 2$, it approaches the vertical asymptote from above.
6. For $x > 2$, the function continues to decrease but switches to concave up, approaching infinity as $x \to \infty$.

This setup allows for a rough graph showing the behavior of the function across the different intervals, with particular focus on concavity and increasing/decreasing behavior.

Would you like more details or assistance with graphing software?

Here are some relative questions:

1. What is the significance of the concavity in determining the shape of the graph?
2. How do vertical asymptotes affect the overall graph behavior?
3. What other information can be inferred from the second derivative test?
4. How can horizontal tangents help in identifying local maxima or minima?
5. What other features, such as inflection points, could influence the final sketch?

Tip: To sketch concave-up and concave-down portions accurately, focus on where the second derivative changes sign.