Relative Extrema of Polynomials Quick Check

Question

3 of 53 of 5 Items

Question
Use the image to answer the question.

A horizontal S-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 4 to 4 in increments of 0.5.

Brandon was asked to identify the relative maximum of the polynomial function h(x) . Which answer choice identifies the correct value(s)?

(1 point)
Responses

There is no relative maximum in this graph.
There is no relative maximum in this graph.

There is a relative maximum at x=0.
There is a relative maximum at x equals 0 .

There is a relative minimum at x=1.
There is a relative minimum at x equals 1 .

There is a relative maximum at x=2.

Answer 1

To determine the relative maximum of the polynomial function \( h(x) \) based on the description of the graph, let's analyze the options provided:

1. There is no relative maximum in this graph.
2. There is a relative maximum at \( x = 0 \).
3. There is a relative minimum at \( x = 1 \).
4. There is a relative maximum at \( x = 2 \).

Given that the curve is described as being horizontal and S-shaped, we can infer that this graph likely has regions where it peaks (relative maxima) and valleys (relative minima).

• A relative maximum occurs where the curve reaches a peak.
• A relative minimum occurs where the curve reaches a valley.

If we assume that at \( x = 0 \) the function reaches a peak before declining toward \( x = 1 \) (where a valley is located), then the choice "There is a relative maximum at \( x = 0 \)" could be correct.

Alternatively, if the maximum occurs at another point, such as at \( x = 2 \), then that would be the right answer.

However, without viewing the actual graph, I can't determine the exact points of relative maximum and minimum. To select the correct answer based on the properties of typical polynomial graphs:

• If there is indeed a peak in the S-shape at either \( x = 0 \) or \( x = 2 \), then choose the corresponding option.
• If the curve does not appear to reach a peak, select the option stating there is no relative maximum.

If you can provide any additional context or a description of the graph, I can assist further in determining the correct answer.