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Which of the following properly describes the relationship between the degree of a polynomial and the number of relative extrema it has?(1 point)
Responses
The number of relative extrema of a polynomial is, at least, one less than the degree.
The number of relative extrema of a polynomial is, at least, one less than the degree.
The number of relative extrema of a polynomial is always one less than the degree.
The number of relative extrema of a polynomial is always one less than the degree.
The number of relative extrema of a polynomial is equal to the degree.
The number of relative extrema of a polynomial is equal to the degree.
The number of relative extrema of a polynomial is, at most, one less than the degree.
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The correct choice is: "The number of relative extrema of a polynomial is, at most, one less than the degree."
Reason: If f(x) is degree n, f'(x) is degree n−1, so it can have at most n−1 real roots (critical points). Relative extrema can occur only at critical points, so there can be at most n−1 relative extrema.
Reason: If f(x) is degree n, f'(x) is degree n−1, so it can have at most n−1 real roots (critical points). Relative extrema can occur only at critical points, so there can be at most n−1 relative extrema.
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