Problem
ALG-B3-M11-P009 Midpoints
#9
★★★★★ Level 5 of 5
Let \(f\) be continuous, \(f(x+y)+f(x-y)=2f(x)\), \(f(0)=1\), and \(f(2)=5\). Find \(f\).
The midpoint equation gives affinity.
Continuous solutions have the form \(f(x)=ax+b\). From \(f(0)=1\), \(b=1\). From \(f(2)=5\): \(2a+1=5\), so \(a=2\). The answer is \(f(x)=2x+1\).
Jensen without a method hint.