>>760,813,816
u "(x) = -(x/2)(1-x) ならば
u '(x) = u '(0) - (1/4)x^2 +(1/6)x^3
u(x) = u(0) + xu '(0) -(1/12)x^3 ＋(1/24)x^4.
u(1) = u(0) + u '(0) -1/12 +1/24 より u '(0)=1/24.
u(x) = u(0) + (1/24)x(1-x){1+x(1-x)} = u(0) + (1/12)|u "|(1+2|u "|).
|u "| = (1/2)x(1-x) = {1-(1-2x)^2}/8 ≦1/8, u(0) ≦ u(x) ≦ u(0) + (1/12)(1/8)(5/4) = u(0) + 5/384.

u "(x)<0 だから上に凸。 x=1/2 で最大値 u(0)+5/384 をとるだろうな。。。