## It is known that the quadratic function f (x) = ax ^ 2 + BX (a, B are constants and a ≠ 0) satisfies the following conditions: F (2) = 0 and the equation f (x) = x has two equal real roots Find the maximum and minimum values of the function on the interval (- 3,3)

f(x)=-1/2x^2+x

Max 1 / 2 min - 17 / 2

### It is known that the quadratic function f (x) = x ^ 2 + ax + B (a, B are constants), satisfies f (0) = f (1), and the equation f (x) = x has two equal real roots Find the analytical formula of function f (x) When x ∈ [0,4], find the value range of function f (x)

(1) From (0) = f (1), B = a + B + 1, that is, a = - 1, then f (x) = x ^ 2-x + B = x, that is, the equation x ^ 2-2x + B = 0 has two equal real roots, that is, the discriminant △ = (- 2) ^ 2-4b = 4-4b = 0, and B = 1. Therefore, the analytical formula of function f (x) is: F (x) = x ^ 2-x + 1 (2) f (x) = (x-1 / 2) ^ 2 + 3 / 4, then f (x) is in [0,1 / 2]

### Among the following functions, 1. Y = - x is the subtractive function on the interval (0, + ∞) ²+ 4X + 1 2. Y = - 3 / x 3. Y = root x 4. Y = (2 / 3) to the power of X

y=-(x-2) ²+ 5. Where x > 2 is the subtractive function, and where x < 2 is the increasing function, which is inconsistent;

Y = - 3 / X is an increasing function when x > 0, which is inconsistent;

Y = root sign x is an increasing function when x > 0, which is inconsistent;

Y = (2 / 3) ^ x is a subtractive function when x > 0, which is consistent with

Select 4

### Function y = log2 ^ (3-2x-x) ²) The monotone decreasing interval of is

Domain definition

3-2x-x ²> 0

x ²+ 2x-3

### X of function y = (1 / 2) ²- Monotonically decreasing interval of 2X-4 power, X of range function y = 3 ²- Monotonic decreasing interval of 2x + 3 power, range

X of function y = (1 / 2) ²- The monotonic decreasing interval of 2X-4 power, and the value ranges are x respectively ²- Increasing interval of 2X-4 [1, + ∞), (0,32]

X of function y = 3 ²- The monotonic decreasing interval of 2x + 3 power, and the value ranges are x respectively ²- Decreasing interval of 2x + 3 (- ∞, 1], [9, + ∞)

### Find the third power of function y = 4x - x ²- Monotone interval and extremum of 2x

First find the derivative of function y, which is y '= 12x ^ 2-2x-2, and then let y' = 0; The extreme point is x = - 1 / 3, x = 1 / 2, so y increases monotonically on (- ∞, - 1 / 3), decreases monotonically on (- 1 / 3,1 / 2), and increases monotonically on (1 / 2, ∞). When x = - 1 / 3, there is a maximum, y = 11 / 27, when x = 1 / 2, there is a minimum, y = 3 / 4