The function has the following properties:

lim f(x) = 3,

lim

2 5 2) = 00

1/）=-8

f(0) = 0, f (5) = 0, f' (-4) = 0, f" (0) = 0

f" (0) > 0 where x < -1 and -1 < x < 0

f" (0) < 0 where 0 < x < 4 and. x > 4 for all real number c

Select all true statements.

﻿﻿﻿﻿f(x) has two vertical asymptotes x = -1 and a = 4..

﻿﻿﻿﻿f(x) is increasing on the intervals (-00, -1) U (4, 00)

﻿﻿﻿﻿f(x) has a local maximum when| x = -4

﻿﻿﻿﻿f(x) is concave up on the intervals (-00, -1) U (-1,0)

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.

Explore the properties of a function with given limits, derivatives, concavity, and potential asymptotes. Analyze the behavior of the function across different intervals and determine true statements based on mathematical principles.

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