question 8 of 10\nwhat is the maximum number of possible extreme values for the function,\n$f(x)=x^{3}-7x…

Explanation:

Step1: Find the derivative

The derivative of (f(x)=x^{3}-7x - 6) is (f^\prime(x)=3x^{2}-7).

Step2: Determine the degree of the derivative

The degree of (f^\prime(x)=3x^{2}-7) (a quadratic function) is (n = 2).

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

The number of extreme values of a function (y = f(x)) is at most equal to the number of real - valued roots of (f^\prime(x)=0). For a polynomial function (y = f(x)), the number of extreme values is at most (n-1) where (n) is the degree of (f(x)). Another way: for a polynomial (y = f(x)), the number of extreme values is at most the degree of (f^\prime(x)). Since the degree of (f^\prime(x)) (a quadratic) is (2), the maximum number of extreme values (found by setting (f^\prime(x)=0) and solving for (x); a quadratic equation (ax^{2}+bx + c=0) has at most (2) real roots) is (2).

Answer:

A. 2