(1)估计原抛物线是y = -x² + mx + n
过(0, 2), n = 2, y = -x² + mx +n
过(-1, 0), 0 = -1 -m + 2, m = 1
y = -x² + x + 2 = -(x + 1)(x - 2)
B(2, 0)
(2)
对称轴x = (-1 + 2)/2 = 1/2
令P(1/2, p); 看来D是对称轴与x轴的交点，D(1/2, 0)
(i) C为顶点， CP² = CD²
(1/2 - 0)² + (p - 2)² = 2² + (1/2)²
p = 4, P(1/2, 4)或p = 0, P(1/2, 0)
(ii) D为顶点，DP² = DC²
(p - 0)² = 2² + (1/2)²
p = ±√17/2
P(1/2, ±√17/2)
(iii) P为顶点，PC² = PD²
(1/2 - 0)² + (p - 2)² = p²
p = 17/16, P(1/2, 17/16)
(3) BC的方程为x/2 + y/2 = 1, y = 2 - x
E(e, 2 - e), 0 < e < 2, F(e, -e² + e + 2)
令过EF的直线与x轴交于E'(e, 0)
四边形CDBF的面积 = 梯形COE'F的面积 + 三角形E'BF的面积 - 三角形COD的面积
= (1/2)*(OC + E'F)*OE' + (1/2)*E'B*E'F - (1/2)*OD*OC
= (1/2)(2 - e² + e + 2)*e + (1/2)(2 - e)*(-e² + e + 2) - (1/2)*(1/2)*2
= -e² + 2e + 3/2 = -(e - 1)² + 5/2
e = 1时，面积最大，为5/2, E(1, 1)