question 2 of 10\nwhat is the maximum number of possible extreme values for the function,\n$f(x)=x^{4}+x^{3}…

Explanation:

Step1: Find the derivative of the function

The derivative of (f(x)=x^{4}+x^{3}-7x^{2}-x + 6) is (f^\prime(x)=4x^{3}+3x^{2}-14x - 1) using the power rule ((x^n)^\prime=nx^{n - 1}).

Step2: Determine the degree of the derivative

The degree of the polynomial (f^\prime(x)=4x^{3}+3x^{2}-14x - 1) is (n = 3).

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

The number of extreme values of a function (y = f(x)) is at most the number of real - roots of its derivative (f^\prime(x)). 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)). Also, the number of real roots of a polynomial (P(x)=a_mx^m+\cdots+a_1x + a_0) is at most (m) (by the fundamental theorem of algebra). The degree of (f^\prime(x)) (a cubic polynomial) is (3), and the number of real roots of (f^\prime(x)) (which gives the critical points of (f(x))) is at most (3).

Answer:

D. 3