Carmen was asked to identify the relative extrema of the polynomial function h(x)

Question

. Which of the following statements should she make?
grid points (0,0) (0.5,1.75) (2,0) (3,3)
(1 point)
Responses

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

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
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 .

Answer 1

To determine the correct statements about the relative and absolute extrema of the polynomial function $h(x)$, we need to analyze the given grid points.

The grid points provided are:

• (0, 0)
• (0.5, 1.75)
• (2, 0)
• (3, 3)

Evaluation of Extrema:

1. At $x=0$, $h(0)=0$.
2. At $x=0.5$, $h(0.5)=1.75$.
3. At $x=2$, $h(2)=0$.
4. At $x=3$, $h(3)=3$.

Absolute Extrema:

• The lowest value occurs at both $x=0$ and $x=2$ with $h(0)=0$ and $h(2)=0$. Therefore, there is no absolute minimum, as they share the same value.
• The highest value occurs at $x=3$ with $h(3)=3$, so there is an absolute maximum at $x=3$.

Relative Extrema:

• At $x=0.5$, the function increases from $(0,0)$ to $(0.5,1.75)$ and then decreases to $(2,0)$, indicating that this point is a relative maximum.
• At $x=2$, it decreases from $(0.5,1.75)$ to $(2,0)$ and then increases to $(3,3)$, indicating that this point is a relative minimum.

Conclusion:

• There is a relative maximum at $x=0.5$.
• There is a relative minimum at $x=2$.
• There is an absolute maximum at $x=3$.

Based on this analysis, the statements Carmen should make include:

• There is a relative maximum at $x=0.5$.
• There is a relative minimum at $x=2$.

Note that there is no absolute minimum at $x=0.5$.

Therefore, the correct response for extremum analysis is:

• There is a relative maximum at $x=0.5$ and a relative minimum at $x=2$.