Let's analyze each statement in the image and determine if they are true or false.

1. Every differentiable function on the interval $[-2, 1]$ must have both a global maximum and a global minimum.
   Answer: True
   Explanation: Since the interval $[-2, 1]$ is closed and bounded, and every differentiable function is continuous, the Extreme Value Theorem guarantees that such a function will have both a maximum and a minimum on this interval.

2. Every function on the interval $(-1, 3]$ must have both a global maximum and a global minimum.
   Answer: False
   Explanation: The interval $(-1, 3]$ is not closed (it’s open on one end), which means a function may not attain a global maximum or minimum on this interval. For instance, $f(x) = \frac{1}{x+1}$ is undefined at $x = -1$ and does not attain a global minimum.

3. Every differentiable function on the interval $(0, 3]$ must have a global minimum.
   Answer: False
   Explanation: The interval $(0, 3]$ is open on the left, so a function may approach a minimum but not attain it. For example, $f(x) = \frac{1}{x}$ on $(0, 3]$ approaches zero as $x$ approaches $3$, but there is no minimum value as $x \to 0^+$.

4. Every continuous function on the interval $[-2, 2]$ must have a global maximum.
   Answer: True
   Explanation: Since $[-2, 2]$ is a closed and bounded interval, the Extreme Value Theorem guarantees that a continuous function will have both a maximum and minimum on this interval.

Would you like further details or explanations on any of these statements?

Here are five related questions to deepen understanding:

1. What is the Extreme Value Theorem, and how does it apply to closed intervals?
2. How does differentiability impact the behavior of a function on open versus closed intervals?
3. Can you provide an example of a function that lacks a global minimum on an open interval?
4. What properties of continuous functions guarantee the existence of maximum and minimum values on closed intervals?
5. How would the answers change if the intervals were open instead of closed?

Tip: When analyzing the existence of global extrema, always check if the interval is closed and bounded. This is crucial for applying the Extreme Value Theorem.