Problem
ALG-B3-M02-P014 A broad answer
#14
★★★★★ Level 5 of 5
Find all \(f:\mathbb R\to\mathbb R\) such that \(f(x+y)=f(x)+f(y)+xy\) for all \(x,y\in\mathbb R\).
Hint. Subtract \(\frac{x^2}{2}\), but do not assume the additive part is linear.
Let \(A(x)=f(x)-\frac{x^2}{2}\). Then \(A(x+y)=A(x)+A(y)\). Therefore all solutions are \(f(x)=\frac{x^2}{2}+A(x)\), where \(A:\mathbb R\to\mathbb R\) is any additive function. Verification follows from \((x+y)^2/2=x^2/2+y^2/2+xy\).
Goal: choose the right first substitution and then verify the found function.