Question

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

Explanation:

Step1: Find the derivative of the function

The derivative of \(f(x)=x^{3}-7x - 6\) is \(f^{\prime}(x)=3x^{2}-7\) using the power rule \((x^{n})^\prime=nx^{n - 1}\).

Step2: Determine the degree of the derivative

The degree of the polynomial \(y = f^{\prime}(x)=3x^{2}-7\) is \(n = 2\).

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

For a function \(y = f(x)\), the number of extreme values is at most the number of real - valued roots of \(f^{\prime}(x)=0\). A polynomial of degree \(n\) has at most \(n\) real roots. Since \(f^{\prime}(x)\) is a quadratic polynomial (\(n = 2\)), the equation \(f^{\prime}(x)=0\) (i.e., \(3x^{2}-7=0\)) has at most \(2\) real roots.

Answer:

D. 2