Answer:
The maximum number of possible extreme values for a fourth-degree polynomial function like f(x) = x^4 + x^3 - 7x^2 - x + 6 is 3.
To determine the number of extreme values, we can find the derivative of the function f(x) and set it equal to zero to solve for critical points.
f(x) = x^4 + x^3 - 7x^2 - x + 6
f'(x) = 4x^3 + 3x^2 - 14x - 1
Setting f'(x) = 0, we can solve for critical points:
4x^3 + 3x^2 - 14x - 1 = 0
Using numerical methods like the cubic formula or numerical approximation techniques, we can find that there are three real roots for this equation, which correspond to the critical points of f(x).
Since f(x) is a fourth-degree polynomial, we know that it has at most four critical points. Therefore, the maximum number of extreme values for f(x) is three, which can be achieved if the function has two local maxima and one local minimum or one local maximum and two local minima.