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A W-shaped curve is plotted on a coordinate plane with the x-axis ranging from negative 2 to 4 in increments of 0.5 and the y-axis ranging from negative 3 to 4 in increments of 0.5.

Carmen was asked to identify the relative extrema of the polynomial function h(x) . Which of the following statements should she make?

(1 point)
Responses

There is a relative maximum at x=0.5.
There is a relative maximum at x equals 0.5 .

There is an absolute minimum at x=0.5.
There is an absolute minimum at x equals 0.5 .

There is an absolute minimum at x=0.5 and a relative minimum at x=2.
There is an absolute minimum at x equals 0.5 and a relative minimum at x equals 2 .

There is an absolute maximum at x=0.5.

1 answer

Based on the description of the W-shaped curve, there are certain key points regarding relative and absolute extrema.

  1. A relative maximum occurs at a peak of the curve.
  2. A relative minimum occurs at a valley of the curve.
  3. An absolute minimum is the lowest point on the function over the specified interval.
  4. An absolute maximum is the highest point on the function over the specified interval.

From the given information, it is likely that:

  • Since it is W-shaped, there will be a relative maximum around the local peaks and relative minima around the local valleys.
  • The lowest point in the range will be considered the absolute minimum.

Given the statements provided, it would be reasonable to conclude:

  • If there is a peak at \( x = 0.5 \), you might say there is a relative maximum at that point.
  • If \( x = 0.5 \) isn't the overall lowest point (i.e., there might be a lower point elsewhere), avoid claiming an absolute minimum at that point unless it is confirmed.
  • If \( x = 2 \) is a local minimum, it would be claimed as a relative minimum.

Based on these points, the most likely correct statement based on the W-shaped curve is:

There is a relative maximum at \( x = 0.5 \).