Question

identify the extrema of the graph.

Explanation:

Step1: Understand the definitions

• Absolute maximum: The highest point on the entire graph.
• Absolute minimum: The lowest point on the entire graph.
• Relative maximum: A point that is higher than the neighboring points.
• Relative minimum: A point that is lower than the neighboring points.

Step2: Analyze each point

• Point A: It is a local high - point (higher than its immediate neighbors), so it is a relative maximum.
• Point B: It is a local low - point (lower than its immediate neighbors), so it is a relative minimum.
• Point C: It is the lowest point on the entire graph shown, so it is the absolute minimum.
• Point D: It is a local high - point (higher than its immediate neighbors), but not the highest on the entire graph.

Answer:

• Absolute Minimum: Point C
• Absolute Maximum: There is none (as the graph may extend further down on the left and we are only given a part of the graph, but within the visible part, the highest \(y\) - value is at point A, but if we consider the domain of the function as the given graph, and since the left - hand side of the graph goes down infinitely (assuming the function is defined for \(x\lt - 3\) as per the axis labels), there is no absolute maximum. If we assume the domain is the \(x\) - values from the left - most point (where the graph starts rising) to \(x = 4\), still point A is not an absolute maximum as the function could have higher values outside the visible window. But if we consider the given plotted points, and assume the domain is the \(x\) - values for which the function is plotted (from the left - most point to \(x = 4\)), since the left - hand side of the graph (for \(x\lt - 3\)) is going down, and within the plotted \(x\) - range (\(x\) from left - most to \(x = 4\)), the \(y\) - value at point A is higher than other non - infinite \(y\) - values. But in a strict calculus sense (if the function is defined for all real \(x\) where it is graphed), since the left - hand side goes down infinitely, there is no absolute maximum. If we consider the finite \(x\) range of the plotted graph (say from \(x=-3\) to \(x = 4\)), point A is a relative maximum. But generally, for the given graph (assuming the function is defined as \(y = f(x)\) for \(x\) in the visible \(x\) - axis range), there is no absolute maximum.
• Relative Minimum: Point B
• Relative Maximum: Point A