Given the function y = ax ^ 3 + BX ^ 2, when x = 1, there is a maximum of 3. Find (1) the analytic expression of the function and write his monotone interval (2) to find the maximum and minimum of the function on [- 2,1]

Given the function y = ax ^ 3 + BX ^ 2, when x = 1, there is a maximum of 3. Find (1) the analytic expression of the function and write his monotone interval (2) to find the maximum and minimum of the function on [- 2,1]

X = 1, there is a maximum, so y '= 3ax ^ 2 + 2bx is 0 when x = 1, that is, 3A + 2B = 0 and the maximum is 3, a + B = 3. To solve the equations a = - 6, B = 9, the analytical formula is y = - 6x ^ 3 + 9x ^ 2Y' = - 18 (x-1) (x + 1), and the extreme point is x = - 1,1, so the maximum and minimum values can only appear at x = - 1,1, - 2