Let's break down each part of this problem:

Given information about $f(x)$:

1. $f(x)$ is continuous everywhere except at $x = -3$, where it has a vertical asymptote.
2. End behavior:
   • $\lim_{x \to -\infty} f(x) = 2$
   • $\lim_{x \to \infty} f(x) = -4$
3. Continuity and differentiability:
   • $f(x)$ is continuous but not differentiable at $x = 5$.
4. First derivative information ($f'(x)$):
   • $f'(x) > 0$ on $(1, 5)$, meaning $f(x)$ is increasing on this interval.
   • $f'(x) < 0$ on $(-\infty, -3)$, $(-3, 1)$, and $(5, \infty)$, meaning $f(x)$ is decreasing on these intervals.
5. Second derivative information ($f''(x)$):
   • $f''(x) > 0$ on $(-3, 5)$ and $(5, \infty)$, indicating concave up on these intervals.
   • $f''(x) < 0$ on $(-\infty, -3)$, indicating concave down on this interval.

---

Part (a): Local Maximum and Minimum

To identify local maxima and minima, let's use the first derivative information, $f'(x)$:

• Decreasing on $(-\infty, -3)$, increasing on $(-3, 1)$, and decreasing again on $(1, 5)$, with increasing on $(5, \infty)$.

1. Local Maximum: From the given intervals of increase and decrease:
   • The function changes from increasing to decreasing at $x = 5$, so $f(x)$ has a local maximum at $x = 5$.

2