A cubic curve y = f(x) = ax^3 + bx^2 + cx + d has a local minimum at x = 1 and a local maximum at x = -1 . The tangent to the curve at the point where it crosses the y -axis is the line y = -3x + 2 . The value of y(2) is

Step-by-step solution

The curve crosses the y -axis at x = 0 . The tangent at this point is given by y = -3x + 2 . Thus, f(0) = 2 and f'(0) = -3 . Substituting x = 0 into f(x) = ax^3 + bx^2 + cx + d , we get d = 2 . Differentiating f(x) , we have f'(x) = 3ax^2 + 2bx + c . Substituting x = 0 into f'(x) , we get c = -3 . The curve has local extrema at x = 1 and x = -1 , so f'(1) = 0 and f'(-1) = 0 . f'(1) = 3a(1)^2 + 2b(1) - 3 = 0 3a + 2b = 3 f'(-1) = 3a(-1)^2 + 2b(-1) - 3 = 0 3a - 2b = 3 Adding the two equations gives 6a = 6 a = 1 . Subt