Question

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

Explanation:

Step1: Recall the rule for extreme values

To find the number of extreme values of a function \( f(x) \), we use the derivative. The extreme values occur where the derivative \( f'(x) = 0 \) (critical points), and the number of real critical points gives the maximum number of extreme values. For a polynomial function of degree \( n \), the derivative is a polynomial of degree \( n - 1 \). The maximum number of real roots (critical points) of the derivative is \( n - 1 \), which gives the maximum number of extreme values.

Step2: Determine the degree of the function

The given function is \( f(x)=x^{3}+4x^{2}-3x - 18 \). The highest power of \( x \) is 3, so the degree \( n = 3 \).

Step3: Calculate the maximum number of extreme values

Using the rule, the maximum number of extreme values is \( n - 1 \). Substituting \( n = 3 \), we get \( 3 - 1=2 \).

Answer:

D. 2