A function of the following form would be a good place to start thinking about this.
[latex] y=B\cdot x^2[/latex]
You can observe that functions with x^2 will be squared to give something to x^4, and then the second derivative will bring it back down to an x^2 again, hence looking like the original function again.
Of course, it's a nonlinear problem, so you need to keep looking at other possible solutions, but I haven't thought about it much further than this observation.
I think B=-A/12 works.