Question

question 8 of 10
what is the maximum number of possible extreme values for the function,
$f(x)=x^{3}-7x - 6$?
a 2
b 1
c 3
d 4

Explanation:

Step1: Find the derivative

The derivative of \(f(x)=x^{3}-7x - 6\) is \(f^\prime(x)=3x^{2}-7\).

Step2: Determine the degree of the derivative

The degree of \(f^\prime(x)=3x^{2}-7\) (a quadratic function) is \(n = 2\).

Step3: Use the relationship between the degree of the derivative and extreme values

The number of extreme values of a function \(y = f(x)\) is at most equal to the number of real - valued roots of \(f^\prime(x)=0\). For a polynomial function \(y = f(x)\), the number of extreme values is at most \(n-1\) where \(n\) is the degree of \(f(x)\). Another way: for a polynomial \(y = f(x)\), the number of extreme values is at most the degree of \(f^\prime(x)\). Since the degree of \(f^\prime(x)\) (a quadratic) is \(2\), the maximum number of extreme values (found by setting \(f^\prime(x)=0\) and solving for \(x\); a quadratic equation \(ax^{2}+bx + c=0\) has at most \(2\) real roots) is \(2\).

Answer:

A. 2