<s>[INST] What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$? What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$? I'm especially curious if there is both an algebraic and calculus-based derivation of the solution.
 [/INST] $f(x)=1-x^2$ inserting that gives:
$$
\begin{eqnarray}
x^2 f(x) + f(1-x) &=&x^2(1-x^2) + (1 - (1-x)^2)\\
&=&x^2-x^4 +(1-(1-2x+x^2))\\
&=&x^2-x^4 +(1-1+2x-x^2)\\
&=&x^2-x^4+2x-x^2 \\
&=& 2x -x^4
\end{eqnarray}
$$
as required. But I admit: I had some help...
But you can also write it as
$$
x^4+f(x) x^2 -2x-f(1-x)=0
$$
and try to solve the depressed quartic equation...
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