19. Let the quadratic function be
$$
f(x)=a x^{2}+(2 b+1) x-a-2(a, b \in \mathbf{R}, a \neq 0)
$$

have at least one root in $[3,4]$. Find the minimum value of $a^{2}+b^{2}$.
19. Regarding the given equation as a linear equation in $a$ and $b$:
$$
\left(x^{2}-1\right) a+2 x b+x-2=0 \text {. }
$$

From the fact that the distance from a point $(a, b)$ on the line to the origin is greater than or equal to the distance from the origin to the line, we get:
$$
\sqrt{a^{2}+b^{2}} \geqslant \frac{|x-2|}{\sqrt{\left(x^{2}-1\right)^{2}+(2 x)^{2}}} .
$$

Let $x-2=t(t \in[1,2])$. Then the right-hand side of equation (1) becomes:
$$
\frac{t}{(t+2)^{2}+1}=\frac{1}{t+\frac{5}{t}+4} \geqslant \frac{1}{10} \text {. }
$$

Therefore, when $t=1, a=-\frac{2}{25}, b=-\frac{3}{50}$, $a^{2}+b^{2}$ attains its minimum value $\frac{1}{100}$.