Question

what is the maximum number of possible extreme values for the function, ( f(x) = x^3 + 4x^2 - 3x - 18 )?
a. 3
b. 4
c. 2
d. 1

Explanation:

Step1: Recall the rule for extreme values

The number of extreme values of a polynomial function is at most one less than its degree. The degree of a polynomial is the highest power of \(x\) in the function.
For the function \(f(x)=x^{3}+4x^{2}-3x - 18\), the degree is \(n = 3\) (since the highest power of \(x\) is \(3\) from the term \(x^{3}\)).

Step2: Calculate the maximum number of extreme values

Using the rule that the maximum number of extreme values is \(n - 1\), where \(n\) is the degree of the polynomial.
Substitute \(n=3\) into \(n - 1\), we get \(3- 1=2\).

Answer:

C. 2