https://file.aiquickdraw.com/mathbot/file/1734499836033u2gicwny.jpg

Let's analyze each of the statements carefully and determine whether they are **True (T)** or **False (F)**.

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### 1. *Every differentiable function on the interval \((-3, 1]\) must have both a maximum and a minimum.*

- The interval \((-3, 1]\) is **half-open** because it excludes \(-3\) and includes \(1\).
- The **Extreme Value Theorem (EVT)** states that a continuous function on a **closed interval** has both a maximum and a minimum.
- Differentiability implies continuity, but the interval here is not fully closed (missing \(-3\)). Therefore, a differentiable function may not achieve its extreme values near \(-3\).

**Answer**: **False**

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### 2. *Every differentiable function on the interval \((-4, -1)\) must have a minimum.*

- The interval \((-4, -1)\) is **open**, meaning it does not include its endpoints.
- On an open interval, there is no guarantee that a differentiable function attains a maximum or minimum because the extreme values could occur at the endpoints, which are excluded.

**Answer**: **False**

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### 3. *Every function on the interval \([0, 2)\) must have both a maximum and a minimum.*

- The interval \([0, 2)\) is **half-closed** (includes \(0\) but excludes \(2\)).
- The **EVT** does not apply to half-closed intervals. Even for continuous functions, there is no guarantee that a minimum or maximum exists, as a function could approach extreme values near \(2\) without reaching them.

**Answer**: **False**

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### 4. *Every continuous function on the interval \([3, 4]\) must have a maximum.*

- The interval \([3, 4]\) is **closed and bounded**.
- By the **Extreme Value Theorem**, a continuous function on a closed and bounded interval attains both a maximum and a minimum.

**Answer**: **True**

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### Final Answers:
1. **F**  
2. **F**  
3. **F**  
4. **T**

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### Would you like a detailed explanation for any of these statements?

### Related Questions:
1. What is the **Extreme Value Theorem** and how is it applied?
2. How does differentiability relate to continuity?
3. Why does the openness of an interval matter for finding extreme values?
4. Can a continuous function on a half-closed interval always achieve its extrema?
5. What happens to extreme values if the function is **not continuous**?

### Tip:
Always verify the type of interval (open, closed, or half-closed) when determining if extreme values exist for a function.

Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false:

1. Every differentiable function on the interval (-3, 1] must have both a maximum and a minimum.
2. Every differentiable function on the interval (-4, -1) must have a minimum.
3. Every function on the interval [0, 2) must have both a maximum and a minimum.
4. Every continuous function on the interval [3, 4] must have a maximum.

Analyze whether statements involving differentiable and continuous functions on open, closed, and half-open intervals hold true. This problem involves key calculus concepts such as the Extreme Value Theorem, differentiability, and continuity, suitable for advanced high school or undergraduate students.

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