It is known that f (x) is a quadratic function, f (- 1) = 2, f ′ (0) = 0, ∫ 01F (x) DX = - 2. (1) find the analytic expression of F (x); (2) find the maximum and minimum of F (x) on [- 1,1]

It is known that f (x) is a quadratic function, f (- 1) = 2, f ′ (0) = 0, ∫ 01F (x) DX = - 2. (1) find the analytic expression of F (x); (2) find the maximum and minimum of F (x) on [- 1,1]

(1) Let f (x) = AX2 + BX + C (a ≠ 0), then f ′ (x) = 2aX + B. from F (- 1) = 2, f ′ (0) = 0, we get a {B + 2 = 0b = 0, that is, C = 2 (x) = 2 (x) = AX2 + BX + BX + C (a) Let f (x) = AX2 + BX + C (a) Let f (x) = 2 + BX = 2aX + B. from F (- 1) = 2, f (- 1) = 2, f (1) = 2, f (x) and ∫ 01F (x) dxdx) DX = ∫ 01 [AX2 + (2 (2-2-a)] [2 (2 + (2-2-2-a-2-2-a (2-A)] [2-2-2-2-2-2-a (2-2-a)]] DX] DX [13 so when x = 0, f (x) min = - 4; When x = ± 1, f (x) max = 2