6. 11. Let $f(x)=a x^{2}+b x+c$.

Suppose the coordinates of $A, B$ are $\left(x_{1}, 0\right),\left(x_{2}, 0\right)$. Then $-10$ has $b>2 \sqrt{a c}, x_{1} x_{2}=\frac{c}{a}0, b2 \sqrt{a c}+1, \sqrt{a}-\sqrt{c}>1, a>$ $(\sqrt{c}+1)^{2} \geqslant(\sqrt{1}+1)^{2}=4$, so $a \geqslant 5 . b>2 \sqrt{a c} \geqslant 2 \sqrt{5}>4, b \geqslant 5$. Therefore, $a+b+c \geqslant 11$, and the quadratic function $f(x)=5 x^{2}+5 x+1$ satisfies the conditions.
6. Given that $a, b, c$ are all positive integers, and the parabola $y=a x^{2}+b x+c$ intersects the $x$-axis at two distinct points $A, B$. If the distances from $A, B$ to the origin are both less than 1. Then the minimum value of $a+b+c$ is $\qquad$ .