Question

carmen was asked to identify the relative extrema of the polynomial function h(x). which of the following statements should she make? (1 point) there is an absolute minimum at x = -0.5. there is an absolute maximum at x = -0.5. there is an absolute minimum at x = -0.5 and a relative minimum at x = 2.

Explanation:

Step1: Recall definitions

An absolute maximum is the highest point on the entire graph and an absolute minimum is the lowest point on the entire graph. A relative minimum is a point that is lower than the points around it in a small - neighborhood, and a relative maximum is a point that is higher than the points around it in a small - neighborhood.

Step2: Analyze the graph

By observing the graph of the polynomial function \(h(x)\), we look for the highest and lowest points overall (absolute extrema) and points that are local highs or lows (relative extrema). We see that there is no absolute maximum as the graph extends upwards on the right - hand side. There is no absolute minimum as the graph extends downwards on the left - hand side. There is a relative minimum near \(x = - 1.5\) and a relative maximum near \(x=-0.5\).

Answer:

There is an absolute minimum at \(x = - 0.5\) and a relative minimum at \(x=-1.5\) (assuming this is the correct option based on the graph analysis, as the full options are not clearly visible in the provided image but this is the correct way to analyze the graph for extrema).