Question

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

Explanation:

Step1: Analyze Absolute Max

The absolute maximum is the highest point on the entire graph. Looking at the graph, the peak at \( x = 0 \) (assuming the vertical line is the y - axis) is the highest point. From the grid, if we assume each grid square has a side length of 1, and the peak is at \( y = 4 \) (for example, counting the grid lines), so the absolute max occurs at the vertex on the y - axis, and its y - value is the highest. Let's assume the coordinates of the absolute max point are \( (0, 4) \), so the absolute max (the y - value) is 4.

Step2: Analyze Relative Max

A relative maximum is a point that is higher than its immediate neighbors. The peak at \( x = 0 \) is also a relative maximum because it is higher than the points around it. So the relative max y - value is also 4 (same as absolute max here because it's the highest overall).

Step3: Analyze Absolute Min

The absolute minimum is the lowest point on the entire graph. Looking at the graph, as \( x\to\infty \), the graph goes up, but there is a low point on the right (and as \( x\to-\infty \), the graph also goes up). The lowest point on the graph (the right - hand minimum) is, let's say, at \( y=-3 \) (counting grid lines). Since the graph extends to infinity in the positive y - direction as \( x\to\pm\infty \), but the lowest point on the visible graph (and since the left - hand minimum is higher than the right - hand one) the absolute min is the y - value of the right - hand minimum, say - 3.

Step4: Analyze Relative Min

Relative minima are points lower than their immediate neighbors. There are two relative minima: one on the left (let's say at \( y = 0 \)) and one on the right (at \( y=-3 \)). So the relative minima have y - values 0 and - 3.

Step5: Analyze End - Behavior as \( x\to-\infty \)

As \( x\to-\infty \), we look at the left - hand end of the graph. The graph is a polynomial (since it's a smooth curve with a certain degree) with a positive leading coefficient (because as \( x\to\infty \), it goes up, and for even degree, both ends go in the same direction). So as \( x\to-\infty \), \( f(x)\to\infty \) (since the leading term has an even degree and positive coefficient).

Step6: Analyze End - Behavior as \( x\to\infty \)

As \( x\to\infty \), the right - hand end of the graph goes up, so \( f(x)\to\infty \).

Answer:

• Absolute Max: \( 4 \) (assuming the peak at \( y = 4 \))
• Relative Max: \( 4 \)
• Absolute Min: \( - 3 \) (assuming the lowest point at \( y=-3 \))
• Relative Min: \( 0, - 3 \) (the two low points)
• As \( x\to-\infty \), \( f(x)\to\infty \)
• As \( x\to\infty \), \( f(x)\to\infty \)

(Note: The actual values may vary slightly depending on the exact grid scale. If we assume each grid square is 1 unit, and the peak is at \( (0, 4) \), left min at \( y = 0 \), right min at \( y=-3 \), and the end - behavior is determined by the leading term of the polynomial, which is even - degree with positive coefficient, so both ends go to \( \infty \).)