Relative Extrema of Polynomials Quick Check

Question

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Question
Use the image to answer the question.

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 maximum at x=0.5.
There is an absolute 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.

Answer 1

To determine which statements are correct regarding the relative extrema of the polynomial function \( h(x) \), we need to identify the locations of the maxima and minima based on the description of the W-shaped curve.

1. Relative Maximum: This is where the function reaches a peak within a certain interval. If \( h(x) \) has a maximum at \( x = 0.5 \), then this would be a relative maximum.

2. Absolute Maximum: This refers to the highest point over the entire interval considered. If \( h(x) \) does not exceed the value at \( x = 0.5 \) anywhere else on the interval, it would be the absolute maximum.

3. Relative Minimum: This is where the function reaches a trough within a certain interval. If there is a minimum point at \( x = 2 \), it would be a relative minimum.

4. Absolute Minimum: This means it is the lowest point over the entire interval considered.

Given that Carmen observed a W-shaped polynomial, she would likely have identified one or more peaks and troughs.

Based on the potential options:

• If \( h(0.5) \) is a peak within the interval, then “There is a relative maximum at \( x = 0.5 \)” is correct.
• If \( h(x) \) at \( x = 0.5 \) is the highest point in the entire range (from -2 to 4), then “There is an absolute maximum at \( x = 0.5 \)” would also be valid.
• If \( h(x) \) at \( x = 2 \) is lower than at \( x = 0.5\) and it is a trough (a low point of the W), then "There is a relative minimum at \( x = 2 \)" is also valid.
• The statements about absolute minimums would apply if there is any point lower than the values at \( x = 2 \) or \( x = 0.5 \) over the entire range.

Without the actual image or specific values, we can't definitively choose one or multiple statements. However, if \( x = 0.5 \) is noted as both the highest point within a smaller interval and/or the highest point overall, then it may be identified as both a relative and absolute maximum. Similarly, any identified trough at \( x = 2 \) would confirm its status as a relative minimum.

In summary, if Carmen observed a high point at \( x = 0.5 \) and a low point (relative minimum) at \( x = 2 \), a potential correct answer could be: "There is an absolute minimum at \( x=0.5 \) and a relative minimum at \( x=2\)."

Please ensure to analyze the graph for confirmation.