Question

1. absolute max: box relative max: box absolute min: box relative min: box as ( x \to -infty ), box as ( x \to infty ), box

Explanation:

Step1: Analyze Absolute Max

Absolute maximum is the highest point on the entire graph. Looking at the graph, as \( x \to -\infty \), the graph goes to \( +\infty \), so there is no absolute maximum (or it's \( \infty \)). But if we consider the visible peaks, wait, no—when \( x \to -\infty \), the function rises without bound. So Absolute Max: Does not exist (or \( \infty \), but typically we say DNE if it's unbounded above).

Step2: Analyze Relative Max

Relative maxima are local peaks. The graph has two local peaks (the "hills"). So we identify their \( y \)-values (but since we need to describe, or note the behavior). Wait, maybe the question is about the end behavior and extrema. Let's re-express:

- Absolute Max: Since as \( x \to -\infty \), \( f(x) \to +\infty \), there's no absolute maximum (unbounded above).
- Relative Max: The two local maximum points (the peaks) have \( y \)-values (we can see two peaks, so relative maxima at those \( x \)-values, but the question might expect the end behavior and extrema nature.
- Absolute Min: The graph has a lowest point? Wait, as \( x \to \infty \), \( f(x) \to -\infty \), so no absolute minimum (unbounded below).
- Relative Min: The two local minimum points (the "valleys").
- As \( x \to -\infty \): The graph goes up, so \( f(x) \to +\infty \).
- As \( x \to \infty \): The graph goes down, so \( f(x) \to -\infty \).

Wait, maybe the question is to fill in the blanks with the behavior or existence. Let's clarify:

1. Absolute Max: Does not exist (because the function increases without bound as \( x \to -\infty \)).
2. Relative Max: The two local maximums (the peaks) – but since we need to state, maybe the \( y \)-values, but from the graph, we can see two peaks, so relative maxima at those points.
3. Absolute Min: Does not exist (because the function decreases without bound as \( x \to \infty \)).
4. Relative Min: The two local minimums (the valleys) – the left valley, the middle valley.
5. As \( x \to -\infty \): \( f(x) \to +\infty \)
6. As \( x \to \infty \): \( f(x) \to -\infty \)

But maybe the problem expects the following:

- Absolute Max: None (or \( \infty \))
- Relative Max: The two local maxima (their \( y \)-coordinates, but since it's a grid, maybe we can see, but the graph is given. Wait, maybe the user expects the end behavior and extrema. Let's proceed with standard analysis:

Absolute Maximum:

A function has an absolute maximum if there's a value \( f(c) \) such that \( f(c) \geq f(x) \) for all \( x \) in the domain. Here, as \( x \to -\infty \), \( f(x) \to +\infty \), so there's no absolute maximum (it's unbounded above). So Absolute Max: Does not exist (DNE).

Relative Maximum:

Relative maxima occur where the function changes from increasing to decreasing. The graph has two such points (the two peaks). So Relative Max: The \( y \)-values of these two peaks (but since we can't see exact coordinates, but from the graph, they are local maxima).

Absolute Minimum:

A function has an absolute minimum if there's a value \( f(c) \) such that \( f(c) \leq f(x) \) for all \( x \) in the domain. Here, as \( x \to \infty \), \( f(x) \to -\infty \), so there's no absolute minimum (unbounded below). So Absolute Min: Does not exist (DNE).

Relative Minimum:

Relative minima occur where the function changes from decreasing to increasing. The graph has two such points (the two valleys: left valley, middle valley). So Relative Min: The \( y \)-values of these two valleys.

End Behavior:

- As \( x \to -\infty \): The left end of the graph goes up, so \( f(x) \to +\infty \).
- As \( x \to \infty \): The right end of the graph goes down, so \( f(x) \to -\infty \).

Answer:

• Absolute Max: Does not exist
• Relative Max: Two local maxima (at the peaks)
• Absolute Min: Does not exist
• Relative Min: Two local minima (at the valleys)
• As \( x \to -\infty \): \( f(x) \to +\infty \)
• As \( x \to \infty \): \( f(x) \to -\infty \)

(Note: If numerical values are needed, we'd need exact coordinates, but from the graph's grid, we can infer the behavior as above.)