设fx在x=0附近有界且满足f(x)-1/2f(x/2)=x∧2,求fx.

Question

设fx在x=0附近有界且满足f(x)-1/2f(x/2)=x∧2,求fx.

Answer 1

let
f(x) =ax^2+bx+c
f(x)-(1/2)f(x/2) =x^2
ax^2+bx+c - (1/2)(a(x/2)^2+b(x/2)+c)=x^2
(7a/8)x^2 + (b/4)x + c/2 =x^2
=> 7a/8 = 1
a=8/7
and
b/4=0 =>b=0
and
c/2 =0 =>c=0
ie
f(x) = (8/7)x^2