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Let f (x) = AX2 + BX + C (a ≠ 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) use a to represent B and C respectively; (2) when B · C reaches the minimum, find the monotone interval of the function g (x) = - f (x) · ex
(1) From F (x) = AX2 + BX + C, we get f '(x) = 2aX + B. because the curve y = f (x) passes through the point (0, 2A + 3), so f (0) = C = 2A + 3, and the tangent of the curve y = f (x) at (- 1, f (- 1)) is perpendicular to the Y axis, so f' (- 1) = 0, that is - 2A + B = 0, so B = 2A. (2) from (1), we get BC = 2A (2a + 3) = 4 (a + 34) 2-94, so when a = - 34, BC gets the minimum value of - 94. At this time, there is b = - 32, C = 32 Thus f (x) = - 34x2-32x + 32, f ′ (x) = - 32x-32, G (x) = - f (x) ex = (34x2 + 32x-32) ex, so g ′ (x) = - F ′ (x) ex + (- f (x)) ex = 34 (x2 + 4x) ex, let g '(x) = 0, the solution is X1 = 0, X2 = - 4. When x ∈ (- ∞, - 4), G' (x) > 0, so g (x) is an increasing function on X ∈ (- ∞, - 4); when x ∈ (- 4,0), G '(x) < 0, so g (x) When x ∈ (0, + ∞), G '(x) > 0, so g (x) is an increasing function on X ∈ (0, + ∞). It can be seen that the monotone increasing interval of function g (x) is (- ∞, - 4) and (0, + ∞), and the monotone increasing interval is (- 4, 0)
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