Question

Given that the polynomial f(x) has degree 4, which of the following most accurately describes the number of turning points of f(x)?

Select the correct answer below:
a. The graph of f(a) has at least 5 turning points.
b. The graph of f(x) has at least 4 turning points.
c. The graph of f(a) has at most 5 turning points.
d. The graph of f(x) has at most 3 turning points.
e. The graph of f(x) has at most 4 turning points.
f. The graph of f(a) has at least 3 turning points

Answer 1

Explanation:

The rule for polynomials is that for a polynomial degree n, the graph will have, at most, n - 1 turning points. For us, if we have a 4th degree polynomial, we will have, at most, 3 turning points. D is your answer.

Answer 2

The correct answer is e. The graph of f(x) has at most 4 turning points.

A polynomial of degree n can have at most n-1 turning points. This can be seen by considering the derivative of the polynomial.

The derivative is a polynomial of degree n-1, and the turning points of the original polynomial correspond to the x-intercepts of the derivative.

Since a polynomial of degree n-1 can have at most n-1 x-intercepts, the original polynomial can have at most n-1 turning points.

Therefore, a polynomial of degree 4 can have at most 4-1=3 turning points. The answer is e. The graph of f(x) has at most 4 turning points.