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 an absolute minimum at x=0.5.
There is an absolute minimum 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 .
There is a relative maximum at x=0.5.
There is an absolute minimum at x=0.5.
To identify the relative extrema of a polynomial function, Carmen should consider the critical points and analyze the concavity of the function. If she finds a point where the function changes from increasing to decreasing (relative maximum) or from decreasing to increasing (relative minimum), she can identify it as a relative extremum.
Based on the given options, Carmen should state that there is a relative maximum at x=0.5.
To identify the relative extrema of a polynomial function, Carmen needs to find the critical points by taking the derivative of the function and setting it equal to zero. The critical points will be potential points of maximum or minimum for the function.
Based on the answer choices given:
1) "There is an absolute minimum at x=0.5." - Carmen cannot determine that without further information. Absolute minimums can only be determined by analyzing the entire graph of the function, not just individual points.
2) "There is an absolute maximum at x=0.5." - Again, Carmen cannot determine this without further information. Absolute maximums require analyzing the entire graph of the function.
3) "There is an absolute minimum at x=0.5 and a relative minimum at x=2." - Similarly, Carmen cannot determine the existence of absolute minimas without analyzing the entire function graph.
4) "There is a relative maximum at x=0.5." - This statement is correct. By finding the critical points and evaluating the function at those points, Carmen can determine if there is a relative maximum. If the derivative changes sign from positive to negative at a specific point, it suggests a relative maximum at that point.
Therefore, Carmen should make the statement: "There is a relative maximum at x=0.5."