https://file.aiquickdraw.com/mathbot/file/1727118316639fvmul3gb.jpg

The question asks which of the following statements is true regarding the relationship between the functions \( F \) and \( f \), where \( F \) is an antiderivative of \( f \). Let's analyze each option carefully.

### 1. "If \( f' > 0 \) on an interval, then \( F \) is increasing on that interval."
- This is **incorrect**. \( F \) is the antiderivative of \( f \), meaning \( F' = f \). If \( f' > 0 \), it tells us about the concavity of \( f \), not \( F \). For \( F \) to be increasing, we need \( F' = f \) to be greater than 0, but this statement deals with \( f' \), which concerns the second derivative of \( F \).

### 2. "If \( f > 0 \) on an interval, then \( F \) is increasing on that interval."
- This is **correct**. Since \( F' = f \), if \( f > 0 \) on an interval, it means the derivative of \( F \) is positive, so \( F \) is increasing on that interval.

### 3. "If \( f' > 0 \) on an interval, then the graph of \( f \) is concave up on that interval."
- This is **true** but unrelated to \( F \). It talks about \( f \) itself and the fact that if \( f' > 0 \), then \( f \) is concave up, but it does not relate to the relationship between \( F \) and \( f \). Therefore, this is irrelevant to the main question.

### 4. "If \( F > 0 \) on an interval, then \( f' \) is increasing on that interval."
- This is **incorrect**. Just because \( F > 0 \) does not imply anything about \( f' \), as \( F \) being positive only tells us something about the function values of \( F \), not the behavior of its second derivative \( f' \).

### Conclusion:
The correct answer is **Option 2**, "If \( f > 0 \) on an interval, then \( F \) is increasing on that interval."

---

Would you like more details or have any further questions?

Here are 5 related questions you can consider:
1. What does it mean for a function to be concave up or concave down?
2. How is the derivative related to the concavity of a function?
3. What is the relationship between the antiderivative and the original function?
4. How can we determine intervals where a function is increasing or decreasing?
5. What is the second derivative test and how is it applied?

**Tip:** When interpreting derivatives, always remember that the first derivative tells you about the rate of change (increasing or decreasing), while the second derivative informs you about the concavity (concave up or down).

Suppose that F is an antiderivative of f. Which of the following statements is true?
1. If f' > 0 on an interval, then F is increasing on that interval.
2. If f > 0 on an interval, then F is increasing on that interval.
3. If f' > 0 on an interval, then the graph of f is concave up on that interval.
4. If F > 0 on an interval, then f' is increasing on that interval.

Analyze the relationship between an antiderivative F and its function f using derivatives. Learn how to determine if F is increasing, concave up, or how the behavior of f impacts F.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation