Given that the image of the function f (x) = AX3 + bx2 passes through the point m (1,4), the tangent of the curve at the point m is just perpendicular to the straight line x + 9y = 0. (1) find the value of real numbers a and B; (2) if the function f (x) monotonically increases in the interval [M, M + 1], find the value range of M

Given that the image of the function f (x) = AX3 + bx2 passes through the point m (1,4), the tangent of the curve at the point m is just perpendicular to the straight line x + 9y = 0. (1) find the value of real numbers a and B; (2) if the function f (x) monotonically increases in the interval [M, M + 1], find the value range of M

(1) The image of ∵ f (x) = AX3 + bx2 passes through the point m (1,4), ∵ a + B = 4 (1) & nbsp If f '(x) = 3ax2 + 2bx, then f' (1) = 3A + 2B (3 points) from the condition f ′ (1) · (− 19) = − 1, that is 3A + 2B = 9, ② formula (5 points) a = 1, B = 3 (2) f (x) = X3 + 3x2 from the solution of (1) and (2)