Problem
ALG-B3-M10-P002 A Given Value
#2
★★★★☆ Level 4 of 5
Let \(f\) be continuous, \(f(x+y)=f(x)+f(y)+2xy\), and \(f(1)=5\). Find \(f\).
First find the general form.
The quadratic correction gives \(f(x)=x^2+cx\). The condition \(f(1)=5\) gives \(1+c=5\), so \(c=4\). The answer is \(f(x)=x^2+4x\).
The second condition fixes the coefficient.