Question

question 2 of 10 what is the maximum number of possible extreme values for the function, $f(x)=x^{3}+4x^{2}-3x - 18$? a. 1 b. 2 c. 3 d. 4

Explanation:

Step1: Recall derivative - extreme value relation

The extreme values of a function occur at critical points where the derivative is zero or undefined. For a polynomial function, the derivative is always defined.

Step2: Differentiate the function

Given $f(x)=x^{3}+4x^{2}-3x - 18$, its derivative $f'(x)$ using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ is $f'(x)=3x^{2}+8x - 3$.

Step3: Determine the degree of the derivative

The derivative $f'(x)=3x^{2}+8x - 3$ is a quadratic function. The degree of a quadratic function is $n = 2$.

Step4: Recall the number of roots of a polynomial

The number of roots of a non - zero polynomial of degree $n$ is at most $n$. A quadratic function ($n = 2$) has at most 2 roots. Since the critical points of $f(x)$ are the roots of $f'(x)$, the function $f(x)$ has at most 2 extreme values.

Answer:

B. 2