Question

Let g be a polynomial function whose graph has a point of inflection at x=3. Which of the following statements about g must be true?

A. The graph of g has a relative extremum at x=3.
B. The graph of g is tangent to the x-axis at x=3.
C. The graph of the rate of change of g has a relative extremum at x=3.
D. The graph of therate of change of g crosses the x axis at x=3.

Answer 1

A polynomial function with a point of inflection does not necessarily have a relative extremum or cross the x-axis at that point. However, the graph of the rate of change of the function does have a relative extremum at the point of inflection.

Step-by-step explanation:

In order for a polynomial function to have a point of inflection at a specific x-value, the second derivative of the function must be zero at that x-value. Therefore, the first statement is not necessarily true, as a polynomial function can have a point of inflection without having a relative extremum at that point.

The graph of a polynomial function is tangent to the x-axis at a specific x-value if and only if the function has a root (zero) at that x-value. Therefore, the second statement is not necessarily true, as a polynomial function can have a point of inflection without being tangent to the x-axis at that point.

The rate of change of a polynomial function is represented by its derivative. The graph of the derivative of a function has a relative extremum at a specific x-value if and only if the graph of the original function has an inflection point at that x-value. Therefore, the third statement is true.

The graph of the rate of change of a polynomial function crosses the x-axis at a specific x-value if and only if the graph of the original function has a relative extremum at that x-value. Therefore, the fourth statement is not necessarily true, as a polynomial function can have a point of inflection without the rate of change crossing the x-axis at that point.