It is known that the function f (x) = (AX-2) ex has an extreme value at x = 1. (I) find the value of a; (II) find the minimum value of F (x) on [M, M + 1]; (III) prove that for any x1, x2 ∈ [0, 2], there is | f (x1) - f (x2) | ≤ E

It is known that the function f (x) = (AX-2) ex has an extreme value at x = 1. (I) find the value of a; (II) find the minimum value of F (x) on [M, M + 1]; (III) prove that for any x1, x2 ∈ [0, 2], there is | f (x1) - f (x2) | ≤ E

(I) f ′ (x) = AEX + (AX-2) ex = (AX + A-2) ex, from the known f ′ (1) = 0, that is, (2a-2) e = 0, the solution is: a = 1, it is verified that when a = 1, the function f (x) = (X-2) ex has a minimum at x = 1, so a = 1; (II) f (x) = (X-2) ex, f ′ (x) = ex + (X-2) ex

The range of function f (x) = log12 (x2 − 2x + 5) is ()
A. [-2，+∞）B. （-∞，-2]C. （0，1）D. （-∞，2]

Let t = x2-2x + 5, from x2-2x + 5 = (x-1) 2 + 4 ≥ 4, we know that the domain of original function is r, t ≥ 4, then log12t ≤ log124 = − 2, so the domain of original function is (- ∞, - 2]. So the answer is B

If the function y = x2-2x + 3 has a maximum value of 3 and a minimum value of 2 in the interval [0, M], then the value range of M is ()
A. [1，∞）B. [0，2]C. （-∞，2]D. [1，2]

It can be seen from the meaning that the axis of symmetry of the parabola is x = 1, with the opening upward ∵ 0 on the left side of the axis of symmetry ∵ the image on the left side of the axis of symmetry is monotonic decreasing ∵ when x = 0 on the left side of the axis of symmetry, there is a maximum value of 3 ∵ and a minimum value of 2 on [0, M]. When x = 1, the value range of y = 2 ∵ m must be greater than or equal to 1 ∵ the image of the parabola about x = 1 symmetry ∵ m must be ≤ 2, so D is selected

If the hyperbola y2-x2 = 1 and XY − x − y + 1x2 − 3x + 2 = m have a unique common point, then the number of elements in the value set of real number m is ()
A. 2B. 4C. 5D. 6

XY − x − y + 1x2 − 3x + 2 = m can be reduced to: y − 1x − 2 = m (x ≠ 1). It represents the straight line L (excluding the point of x = 1) passing through a (2,1) and with a slope of M. as shown in the figure, let the two intersections of the straight line x = 1 and the hyperbola y2-x2 = 1 be m, n respectively