If y = (M + 1) XM2 − m-3x + 1 is a quadratic function, then the value of M is______ ．

If y = (M + 1) XM2 − m-3x + 1 is a quadratic function, then the value of M is______ ．

The answer is: M + 1 ≠ 0, m ≠ - 1, m2-m = 2, M1 = 2, M2 = - 1

For the cubic function f (x) = x 3-3x 2-3mx + 4 (where m is a constant), answer the following question
(1) find the maximum of F (x); (2) find the value of m when the maximum of F (x) is 5; (3) find the tangent equation of the curve y = f (x) passing through the origin

(1) F '(x) = 3x ^ 2-6x-3m = 0, then x = 1-radical (1 + m) or x = 1 + radical (1 + m)
According to the application of derivative, when x = 1-radical (1 + m), f (x) has maximum value = 2 (M + 1) radical (1 + m) + 2-3m
(2) Through the conclusion of (1): F (1 + m)) = 5, we can get: M = 5 / 4
(3) Let P (a, b), f (a) = a 3-3a 2-3ma + 4 = B@
F '(a) = 3A ^ 2-6a-3m = K (k is the slope of tangent)
The tangent equation is y-b = (3a ^ 2-6a-3m) (x-a). By introducing the origin and @ formula into the tangent equation, we can get a = 2, B = - 6m
The tangent equation is y = - 3mx

The function f (x) = - (1 / 2) x ^ 2-3x - (5 / 2) is known
(1) The monotone interval, range and zero point of the function are obtained;
(2) The values of F (- 1 / 4) and f (- 15 / 4) are compared;
(3) Write the set of x such that f (x) is less than 0

1) Increasing interval: X is less than or equal to 3; decreasing interval: X is greater than or equal to 3; range: y is less than or equal to 2; zero: x = 5, x = 1
2) Because x = - 1 / 4 and x = - 15 / 4 are in the increasing range, f (- 1 / 4) > F (- 15 / 4)
3){x:x5}