Show $a+b\leq\frac{8}{27}$

Question

We have a polynomial $f(x)=x^3-x^2+ax+b$ with $a,b$ real numbers. If $x_{1},x_{2},x_{3}$ are the roots of $f$ and are positive real numbers show that $a+b\leq\frac{8}{27}$. I showed using Viete formulas and identities that $a+b\leq\frac{1}{3}$. But I dont know where the $8$ should come from.

Answer 1

AM-GM!

$2=(1-x_1)+(1-x_2)+(1-x_3) \ge 3 \times (f(1))^\frac13$