1. Let $f(x)=a x^{2}+b x+1$.

a) Determine $f(x+1)-f(x)$.

b) If $f(x+1)-f(x)=x$, find $a$ and $b$.
Solution. We have
a) $f(x+1)-f(x)=a(x+1)^{2}+b(x+1)+1-a x^{2}-b x-1=2 a x+a+b$

b) From $f(x+1)-f(x)=x$ we have $2 a=1$ and $a+b=0$, i.e., $a=\frac{1}{2}$ and $b=-\frac{1}{2}$.