question 9 of 10\nwhat is the maximum number of possible extreme values for the function,\n$f(x)=x^{4}+x^{3}…

Explanation:

Step1: Recall the relationship between the degree of a polynomial and its derivative

The degree of the polynomial (f(x)=x^{4}+x^{3}-7x^{2}-x + 6) is (n = 4). The derivative of a polynomial (y=a_{n}x^{n}+a_{n - 1}x^{n-1}+\cdots+a_{1}x + a_{0}) is given by (y^\prime=na_{n}x^{n - 1}+(n - 1)a_{n-1}x^{n-2}+\cdots+a_{1}). So, the derivative of (f(x)) is (f^\prime(x)=4x^{3}+3x^{2}-14x-1), and its degree is (n-1=3).

Step2: Recall the relationship between the degree of the derivative and the number of extreme values

The number of extreme values of a function (y = f(x)) is at most the number of real - roots of its derivative (y = f^\prime(x)). A polynomial of degree (m) has at most (m) real roots. Since the degree of (f^\prime(x)) is (3), the function (f(x)) has at most (3) extreme values.

Answer:

B. 3