Problem
ALG-B3-M05-P016 The Pure Quadratic Equation
#16
★★★★★ Level 5 of 5
Find all continuous \(f:\mathbb R\to\mathbb R\) such that \(f(x+y)+f(x-y)=2f(x)+2f(y)\) and \(f(2)=12\).
Continuous solutions have the form \(cx^2\).
With \(y=0\), we get \(f(0)=0\). Continuous solutions of the quadratic Cauchy equation have the form \(f(x)=cx^2\). From \(f(2)=12\), \(4c=12\), so \(c=3\). The answer is \(f(x)=3x^2\). Checking follows by expanding squares.
Gives a separate template for the quadratic version.