Relative Extrema of Polynomials Quick Check

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A horizontal S-shaped curve is plotted on a coordinate plane with both axes ranging from negative 5 to 5 in unit increments.

Which of the following most accurately identifies the relative maximum and minimum of the polynomial function h(x) ?

(1 point)
Responses

There is a relative maximum at x=−1 and no relative minimum.
There is a relative maximum at x equals negative 1 and no relative minimum.

There is a relative maximum at x=−1 and a relative minimum at x=0.555.
There is a relative maximum at x equals negative 1 and a relative minimum at x equals 0.555 .

There is a relative maximum at x=1.5 and a relative minimum at x=−2.
There is a relative maximum at x equals 1.5 and a relative minimum at x equals negative 2 .

There is a relative minimum at x=−1 and a relative maximum at x=0.555.
There is a relative minimum at x equals negative 1 and a relative maximum at x equals 0.555 .
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Answer 1

To identify the relative extrema of the polynomial function \( h(x) \) based on the description of the curve (a horizontal S-shaped curve), we can infer a few things about typical S-shaped curves:

1. Relative Maximum and Minimum: An S-shaped curve often has one relative maximum and one relative minimum. The peaks (relative maxima) and valleys (relative minima) are typically where the slope of the function changes from positive to negative or from negative to positive.

2. Location of Extrema: Given your options:

   • The relative maximum at \( x = -1 \) suggests that at this point, the curve is at a peak.
   • The possible relative minimum at \( x = 0.555 \) suggests a valley of the curve.

With the context of a horizontal S-shaped curve and if the S-shape is situated as described, it is reasonable to conclude:

• Option: "There is a relative maximum at \( x = -1 \) and a relative minimum at \( x = 0.555 \)."

This option accurately describes the behavior typical to the described S-shaped curve, indicating the turning points in the function.