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

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