Which of the following properly describes the relationship between the degree of a polynomial and the number of relative extrema it has?
Responses
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. - no response given
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. - incorrect
The number of relative extrema of a polynomial is, at most, one less than the degree.
The number of relative extrema of a polynomial is, at most, one less than the degree. - not selected, this is the correct answer
The number of relative extrema of a polynomial is, at least, one less than the degree.
explain how you get that answer
The correct answer is "The number of relative extrema of a polynomial is, at most, one less than the degree."
To understand why, let's consider the behavior of a polynomial function. The degree of a polynomial determines the highest power of the variable in the expression.
For example, a polynomial of degree 3 will have a third-degree term (x^3) present. The behavior of the polynomial function changes depending on the sign and the coefficient of this highest-degree term.
Now, let's focus on the concept of relative extrema. Relative extrema are the local maxima and minima points on the graph of a function. They occur where the slope of the function changes from positive to negative (a local maximum) or from negative to positive (a local minimum).
The number of relative extrema a polynomial can have is determined by the number of times the sign of the derivative changes. In other words, it depends on the number of times the slope of the function changes from positive to negative or vice versa.
The derivative of a polynomial reveals the slopes at different points on its graph. The degree of the polynomial dictates the highest power of the derivative.
Starting with a polynomial of degree n, the highest power of the derivative will be n-1. This means that the number of times the sign of the derivative can change (and therefore the number of relative extrema) is at most n-1.
For example, a quadratic polynomial (degree 2) can have at most 1 relative extremum (a maximum or a minimum). A cubic polynomial (degree 3) can have at most 2 relative extrema, and so on.
It's important to note that a polynomial function may have fewer relative extrema or even none at all. The relationship described in the correct answer statement ensures that the number of relative extrema is never greater than the degree of the polynomial.