Question

question 3 of 10
what is the maximum number of possible extreme values for the function,
$f(x)=x^{4}+x^{3}-7x^{2}-x + 6$?
a. 3
b. 5
c. 2
d. 4

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