1. Let the function be
$$
\begin{aligned}
f(x)= & \sqrt{4 x^{4}-3 x^{2}-10 x+26}- \\
& \sqrt{4 x^{4}-19 x^{2}-6 x+34} .
\end{aligned}
$$

When $x=$ $\qquad$, the function $y=f(x)$ achieves its maximum value.
$-1 . \frac{-1 \pm \sqrt{23}}{2}$.
Notice that,
$$
\begin{aligned}
f(x)= & \sqrt{(x-5)^{2}+\left(2 x^{2}-1\right)^{2}}- \\
& \sqrt{(x-3)^{2}+\left(2 x^{2}-5\right)^{2}} .
\end{aligned}
$$

Let point $P\left(x, 2 x^{2}\right), A(5,1), B(3,5)$. Then
$$
f(x)=|P A|-|P B| \leqslant|A B| .
$$

When and only when $P, A, B$ are collinear, the equality holds, at this time, $x=\frac{-1 \pm \sqrt{23}}{2}$.