Problem
ALG-B3-M10-P009 Two Given Values
#9
★★★★★ Level 5 of 5
Let \(f\) be continuous, \(f(x+y)=f(x)+f(y)+a xy\), \(f(1)=1\), and \(f(2)=6\). Find \(a\) and \(f\).
Use \(f(x)=\frac a2x^2+cx\).
We have \(f(x)=\frac a2x^2+cx\). From \(f(1)=1\): \(\frac a2+c=1\). From \(f(2)=6\): \(2a+2c=6\), i.e. \(a+c=3\). Subtracting gives \(\frac a2=2\), hence \(a=4\), \(c=-1\). The answer is \(f(x)=2x^2-x\).
The parameter is determined by two values.