Question Number 11517 by Joel576 last updated on 27/Mar/17

f : R → R  If  x^2 f(x) + f(1−x) = 2x − x^4   Determine f(x)

Answered by sma3l2996 last updated on 27/Mar/17

let u=1−x  (1−u)^2 f(1−u)+f(u)=2(1−u)−(1−u)^4   so f(1−u)=((2−(1−u)^3 )/((1−u)))−((f(u))/((1−u)^2 ))  f(1−x)=((2−(1−x)^3 )/((1−x)))−((f(x))/((1−x)^2 ))  x^2 f(x)+f(1−x)=x^2 f(x)−((f(x))/((1−x)^2 ))+((2−(1−x)^3 )/((1−x)))  f(x)(((x^2 (1−x)^2 −1)/((1−x)^2 )))+((2−(1−x)^3 )/((1−x)))=2x−x^4   f(x)=(2x−x^4 +(((1−x)^3 −2)/((1−x))))((((1−x)^2 )/(x^2 (1−x)^2 −1)))

Commented byJoel576 last updated on 28/Mar/17

thank you very much

Commented byJoel576 last updated on 28/Mar/17

one more, how can we determine the value  of f(2009) ?

Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 28/Mar/17

x^2 f(x)+f(1−x)=2x−x^4   (1−x)^2 f(1−x)+f(x)=2(1−x)−(1−x)^4   [x^2 (1−x)^2 −1]f(x)=(2x−x^4 )(1−x)^2 −2(1−x)+(1−x)^4 =  (1−x)[2x−x^4 −2x^2 +x^5 −2+1−3x+3x^2 −x^3 )=  (1−x)(x^5 −x^4 −x^3 +x^2 −x−1)=  −x^6 +x^5 +x^4 −x^3 +x^2 +x+x^5 −x^4 −x^3 +x^2 −x−1=  −(x^6 −2x^5 +2x^3 −2x^2 +1)=  −(x^3 −x^2 +1)^2 +x^4 =(x^2 +x^3 −x^2 +1)(x^2 −x^3 +x^2 −1)=  (x^3 +1)(−x^3 +2x^2 −1)=(x^3 +1)(−x(x^2 −2x+1)−(1−x))=  (x^3 +1)(−x(1−x)^2 −(1−x))  ⇒f(x)=(((1+x)(1−x+x^2 )(1−x)(1+x(1−x)))/((1−x(1−x))(1+x(1−x))))=  f(x)=(1−x)(1+x)=1−x^2   f(2009)=(1−2009)(1+2009)=  −2010×2008