To analyze the properties of the function $f(x)$ based on the given information, let's go through each of the provided statements and determine which are true.

1. $f(x)$ has two vertical asymptotes at $x = -1$ and $x = 4$:

• Asymptotes occur where the function approaches infinity as $x$ approaches certain values. However, the problem statement doesn't provide enough information about $f(x)$ approaching infinity at $x = -1$ and $x = 4$. Thus, this statement cannot be confirmed as true.

2. $f(x)$ is increasing on the intervals $(-\infty, -1) \cup (4, \infty)$:

• The statement doesn't provide enough information about $f'(x)$ (the derivative) on these intervals, so we cannot confirm this statement.

3. $f(x)$ has a local maximum at $x = -4$:

• We are given that $f'(-4) = 0$. This implies that there could be a local maximum, minimum, or a saddle point at $x = -4$. Without knowing the sign of $f''(-4)$, we cannot confirm this statement either.

4. $f(x)$ is concave up on the intervals $(-\infty, -1) \cup (-1, 0)$:

• We are given that $f''(x) > 0$ for $x < -1$ and $-1 < x < 0$. Since $f''(x) > 0$ implies concavity up, this statement is true.

Summary:

Based on the analysis, the only statement that can be confirmed as true is:

• $f(x)$ is concave up on the intervals $(-\infty, -1) \cup (-1, 0)$.

Let me know if you want further details or have any questions.

Here are some related questions to deepen your understanding:

1. What are the conditions for a function to have a vertical asymptote?
2. How can you determine where a function is increasing or decreasing?
3. What additional information do we need to confirm if $f(x)$ has a local maximum at $x = -4$?
4. How does the sign of the second derivative affect the concavity of a function?
5. What is the relationship between the first derivative and the critical points of a function?
6. How would you determine if a point is a point of inflection?
7. What does it mean for a function to be concave down?
8. How do you confirm the existence of a local minimum at a certain point?

Tip: Always examine the first and second derivatives to understand the behavior of a function around critical points.