Problem
ALG-B3-M10-P017 Parametric Jensen Equation
#17
★★★★★ Level 5 of 5
Find all continuous \(f:\mathbb R\to\mathbb R\) such that \(f(x+y)+f(x-y)=2f(x)+2a y^2\).
Subtract \(a x^2\).
Let \(g(x)=f(x)-a x^2\). Then \(g(x+y)+g(x-y)=2g(x)\). Continuous solutions of this equation are affine: \(g(x)=cx+d\). Therefore \(f(x)=a x^2+cx+d\). Direct checking works.
Parametric version of the second-difference equation.