Given that the function g (x) = - x2-3, f (x) is a quadratic function, when x ∈ [- 1,2], the minimum value of F (x) is 1, and f (x) + G (x) is an odd function, the analytic expression of F (x) is obtained

Given that the function g (x) = - x2-3, f (x) is a quadratic function, when x ∈ [- 1,2], the minimum value of F (x) is 1, and f (x) + G (x) is an odd function, the analytic expression of F (x) is obtained

Let f (x) = AX2 + BX + C (a ≠ 0), then f (x) + G (x) = (A-1) x2 + BX + C-3, ∵ f (x) + G (x) is an odd function, ∵ a = 1, C = 3 ∵ f (x) = x2 + BX + 3, symmetry axis X = - B2. ① when - B2 > 2, i.e. B ＜ - 4, f (x) is a decreasing function on [- 1, 2], and the minimum value of ∵ f (x) is f (2) = 4 + 2B + 3 = 1, ∵ B = - 3, ∵ then there is no solution. ② when - 1 ≤ - B2 ≤ 2, i.e. - 4 ≤ B ≤ 2, f (x) Min = f (- B2) = 3-b24 = 1, B = ± 22, B = - 22, then f (x) = x2-22x + 3, ③ when - B2 < - 1s, i.e. b > 2, f (x) is an increasing function on [- 1, 2], and the minimum value of F (x) is f (- 1) = 4-b = 1, B = 3, f (x) = x2 + 3x + 3. In conclusion, f (x) = x2-22x + 3, or F (x) = x2 + 3x + 3