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

question 3 of 10\nwhat is the maximum number of possible extreme values for the function,\n$f(x)=x^{4}+x^{3}-7x^{2}-x + 6$?\na. 3\nb. 5\nc. 2\nd. 4

question 3 of 10\nwhat is the maximum number of possible extreme values for the function,\n$f(x)=x^{4}+x^{3}-7x^{2}-x + 6$?\na. 3\nb. 5\nc. 2\nd. 4

Answer

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)), (f^\prime(x)=4x^{3}+3x^{2}-14x - 1) has degree (n-1=3).

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

The extreme values of a function (y = f(x)) occur at the critical points, where (f^\prime(x)=0). A polynomial equation of degree (m) has at most (m) real - valued solutions. Since the degree of (f^\prime(x)) is (m = 3), the equation (f^\prime(x)=0) has at most (3) real - valued solutions.

Answer:

A. 3