Given that the image of function f (x) = (AX2 + BX + C) E-X passes through a point (0, 2a), and the inclination angle of the tangent at this point is 45 ° (1) B, C are represented by a; (2) if f (x) is a monotone increasing function on [2, + ∞), the value range of a is obtained

Given that the image of function f (x) = (AX2 + BX + C) E-X passes through a point (0, 2a), and the inclination angle of the tangent at this point is 45 ° (1) B, C are represented by a; (2) if f (x) is a monotone increasing function on [2, + ∞), the value range of a is obtained

(1) F ′ (x) = - [AX2 + (b-2a) x + C-B] E-X is known as: F / (0) = B − C = 1F (0) = 2A, C = 2Ab = 1 + 2A (2) f ′ (x) = - (AX2 + x-1) E-X ∵ f (x) is a monotone increasing function on [2, + ∞), then f ′ (x) ≥ 0 is constant on [2, + ∞)

The function f (x) = (ax ＾ 2 + BX + C) e ＾ (- x) (a is not equal to 0) is known, and the tangent equation at this point is 4x-y-2 = 0
1.  if f (x) is a monotone increasing function in [2, + ∞), the value range of real number a is obtained
2.  if the function f (x) = f (x) - M has exactly one zero point, find the value range of real number M

(1) By sorting out the tangent equation, we get that Y - (- 2) = 4 (x-0), the slope of the tangent at (0, - 2) is 4, let x = 0, y = - 2C × e ^ (- 0) = C = - 2F (x) = (ax & # 178;) e ^ (- x) + (BX) e ^ (- x) - 2E ^ (- x) f '(x) = (2aX) e ^ (- x) - (AX & # 178;) e (- x) + B × e ^ (- x) - (BX) e ^ (- x) + 2E e ^ (- x) x = 0

Let f (x) = ax + BX + C (a is not equal to 0), y = f (x) pass through the point (0,2a + 3), and the tangent at the point (- 1, f (- 1)) is perpendicular to the Y axis
1. B and C are represented by A
2. When BC is the minimum, we find the monotone interval of G (x) = - f (x) multiplied by e minus X

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