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:
- At \( x = 0 \), \( h(0) = 0 \).
- At \( x = 0.5 \), \( h(0.5) = 1.75 \).
- At \( x = 2 \), \( h(2) = 0 \).
- 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 \( x = 2 \).