A cubic polynomial function f(x) = x^3 + px^2 + qx + r has a local maximum at x = -1 and a local minimum at x = 3 . If the local minimum value of the function is 10 , then its local maximum value is equal to :

Step-by-step solution

Since f(x) has local extrema at x = -1 and x = 3 , these points must be the roots of f'(x) = 0 . We have f'(x) = 3x^2 + 2px + q . Since the leading coefficient of f(x) is 1 , the leading coefficient of f'(x) is 3 . Thus, we can write f'(x) using its roots: f'(x) = 3(x + 1)(x - 3) f'(x) = 3(x^2 - 2x - 3) = 3x^2 - 6x - 9 Comparing the coefficients with 3x^2 + 2px + q , we get: 2p = -6 p = -3 q = -9 So, the function is f(x) = x^3 - 3x^2 - 9x + r . The local minimum occurs at x = 3 . We are given that the local minimum