Question

1. absolute max:

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

Answer:

To solve this problem, we analyze the graph of the function:

Absolute Max:

The absolute maximum is the highest point on the entire graph. Looking at the graph, the right - most peak is the highest point. Since we assume the grid has a certain scale (let's assume each grid square has a side length of 1 unit for simplicity), if we look at the y - coordinate of the highest point, we need to check the graph. But generally, for a function's graph, the absolute maximum is the maximum value of the function over its entire domain. From the graph, the right - hand peak is the highest, and if we consider the end - behavior (both ends go to $-\infty$), the absolute maximum occurs at the right - most local maximum. Since we don't have the exact equation, but from the graph's shape, the absolute maximum is the y - value of the highest point. However, if we assume the graph's highest point (the right - most peak) and the left - most peak, the right - most peak is higher. But since the problem is about identifying the concepts:

Relative Max:

Relative maxima are the peaks of the graph (local maxima). There are two peaks in the graph. The left - hand peak and the right - hand peak are relative maxima.

Absolute Min:

Since as $x
ightarrow\pm\infty$, the function goes to $-\infty$, the function does not have an absolute minimum (because it can keep decreasing without bound).

Relative Min:

The relative minimum is the valley between the two peaks (the lowest point between the two local maxima).

As $x

ightarrow-\infty$:
Looking at the left - hand side of the graph, as $x$ approaches $-\infty$, the graph is going down, so $f(x)
ightarrow-\infty$.

As $x

ightarrow\infty$:
Looking at the right - hand side of the graph, as $x$ approaches $\infty$, the graph is going down, so $f(x)
ightarrow-\infty$.

Final Answers:

• Absolute Max: The y - value of the highest peak (right - most peak). If we assume the graph's scale, but since it's a graph - based question, we can say that the absolute maximum is the maximum value among the local maxima, and the right - most local maximum is the absolute maximum. But in terms of the answer (assuming we are to describe the behavior or the value, but since the graph is not labeled with exact values, we can say that the absolute maximum exists at the right - most local maximum, and the function does not have an absolute minimum (since it tends to $-\infty$ at both ends), the relative maxima are the two peaks, the relative minimum is the valley between them, and as $x

ightarrow\pm\infty$, $f(x)
ightarrow-\infty$.

But if we are to fill in the blanks (assuming the graph has a standard grid and we can estimate):

• Absolute Max: Does not exist (since the function goes to $-\infty$ at both ends, but wait, no - the function has a highest point. Wait, the graph has two peaks, and the right - hand peak is higher. So the absolute maximum is the y - coordinate of the right - hand peak. If we assume the grid, let's say the right - hand peak is at a higher y - value. But maybe the problem expects the following:

• Absolute Max: The y - value of the highest point (right - most local maximum)
• Relative Max: The y - values of the two peaks (left and right local maxima)
• Absolute Min: Does not exist (since $\lim_{x

ightarrow\pm\infty}f(x)=-\infty$)

• Relative Min: The y - value of the valley between the two peaks
• As $x

ightarrow-\infty$: $f(x)
ightarrow-\infty$

• As $x

ightarrow\infty$: $f(x)
ightarrow-\infty$

If we have to write in the boxes (assuming the graph is on a coordinate system with integer coordinates, and we can see the behavior):

• Absolute Max: The maximum value of the function (the highest point on the graph, e.g., if the right - most peak is at y = 4 (assuming grid), but since the graph is not labeled, we can say that the absolute maximum is the y - coordinate of the right - hand local maximum.
• Relative Max: The y - coordinates of the two local maxima (left and right peaks)
• Absolute Min: Does not exist (because the function decreases without bound as $x

ightarrow\pm\infty$)

• Relative Min: The y - coordinate of the local minimum (the valley)
• As $x

ightarrow-\infty$: $f(x)
ightarrow-\infty$

• As $x

ightarrow\infty$: $f(x)
ightarrow-\infty$

But to be precise, for a function's graph:

1. Absolute Maximum: The highest point on the entire graph. If the graph has a peak that is higher than all others and the function is defined on a domain where it doesn't go to $+\infty$, then that peak is the absolute maximum. In this graph, the right - hand peak is the highest, so the absolute maximum is the y - value of that peak.
2. Relative Maximum: A point where the function changes from increasing to decreasing (a local peak). There are two such points (the two peaks), so their y - values are relative maxima.
3. Absolute Minimum: A function has an absolute minimum if there is a point that is lower than all other points. Since as $x

ightarrow\pm\infty$, the function goes to $-\infty$, there is no absolute minimum (the function can get arbitrarily low).

1. Relative Minimum: A point where the function changes from decreasing to increasing (a local valley). There is one such point (the valley between the two peaks), so its y - value is the relative minimum.
2. **As $x

ightarrow-\infty$**: Looking at the left - hand end of the graph, the function is decreasing, so $f(x)
ightarrow-\infty$.

1. **As $x

ightarrow\infty$**: Looking at the right - hand end of the graph, the function is decreasing, so $f(x)
ightarrow-\infty$.

So, to fill in the blanks (assuming we use the standard definitions):

• Absolute Max: The y - value of the highest peak (right - most local maximum)
• Relative Max: The y - values of the two local maxima (left and right peaks)
• Absolute Min: Does not exist
• Relative Min: The y - value of the local minimum (the valley)
• As $x

ightarrow-\infty$: $f(x)
ightarrow-\infty$

• As $x

ightarrow\infty$: $f(x)
ightarrow-\infty$

If we are to write the answers in the boxes (using the standard notation for end - behavior and extrema):

• Absolute Max: (the y - coordinate of the right - most peak, e.g., if we assume the graph's right - most peak is at a higher y - value, but since it's not labeled, we can say that the absolute maximum is the maximum of the local maxima, and the right - most local maximum is the absolute maximum. But in terms of the answer, if we consider the graph's behavior)
• Relative Max: Two values (the y - coordinates of the two peaks)
• Absolute Min: Does not exist
• Relative Min: One value (the y - coordinate of the valley)
• As $x

ightarrow-\infty$: $-\infty$

• As $x

ightarrow\infty$: $-\infty$

So, for the end - behavior:

As $x
ightarrow-\infty$, $f(x)
ightarrow\boldsymbol{-\infty}$

As $x
ightarrow\infty$, $f(x)
ightarrow\boldsymbol{-\infty}$

For extrema:

• Absolute Max: Exists at the right - most local maximum (its y - value)
• Relative Max: Two values (the y - values of the two local maxima)
• Absolute Min: Does not exist
• Relative Min: One value (the y - value of the local minimum)