抛物线最高点坐标三角形

y = x =4,x = 1
y可表达为y = a(x - 4)(x - 1) = ax^2 -5ax +4a
x = 0,y = 4a = -2
a = -1/2
y = -(x-4)(x-1)/2 = -x^2/2 + 5x/2 -2
D(m,n)
n = -m^2/2 + 5m/2 -2
DC的方程:(y +2)/(x -0) = (n +2)/(m -0)
m(y+2) = (n+2)x
与x轴的交点为E:(2m/(n+2),0)
三角形DCA的面积 = 三角形ECA的面积 + 三角形EDA的面积 (n ≥ 0)
三角形DCA的面积 = 三角形ECA的面积 - 三角形EDA的面积 (n < 0,显然不用考虑)
三角形DEA的面积 = (1/2)|AE|*|D的纵坐标| = (1/2)[4 - 2m/(n+2)]|n| = [2 - m/(n+2)]|n|
三角形ECA的面积 = (1/2)|AE|*|C的纵坐标| = (1/2)[4 - 2m/(n+2)]|-2| = 4 - 2m/(n+2)
三角形DCA的面积S = [2 - m/(n+2)]|n| + 4 - 2m/(n+2)
= n [2 - m/(n+2)] + 4 - 2m/(n+2)
= [2 - m/(n+2)]*(n+2)
= 2(n+2) -m
= 2(-m^2/2 + 5m/2 -2 + 2) -m
= -m^2 + 4m
= -2(m-2)^2 +4
m = 2时,S最大,D(2,1)

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