The function f:R⇒R satisfies xf(x)+f(1-x)=x^3-x for all real x. Find f(x).

Answer 1

xf(x) + f(1-x) = x(x-1)(x+1)
(1-x)f(1-x) + f(1-(1-x)) = (1-x)((1-x)-1)(1-x+1)

multiply the first by (1-x):

x(1-x)f(x) + (1-x)f(1-x) = x(x-1)(x+1)(1-x)
f(x) + (1-x)f(1-x) = (1-x)(-x)(2-x)

subtract to get

(x(1-x)-1)f(x) = (x(x-1)(x+1)(1-x))-((1-x)(-x)(2-x))

f(x) = -x(1-x)(x^2-x+1)/(x^2-x+1) = -x(1-x)

check:

xf(x) + f(1-x)
= x(-x(1-x)) - (1-x)(x)
= -x^2+x^3-x+x^2
= x^3-x