Find the distance from the origin to the closest point(s) on the graph of $$12 x^2 + 14 xy + 12 y^2 = 8$$

We begin by computing $\frac{dy}{dx}$ implicitly for points $(x,y)$ on the graph, obtaining:
$\displaystyle\frac{dy}{dx} =$
We need to minimize the following function of $x$ and $y$:
$f(x,y)=$
There are four critical points, in increasing order:
$x_1 =$
$x_2 =$
$x_3 =$
$x_4 =$
Thus there is/are point(s) closest to the origin, one of which is
$(x,y)= ($ $,$$)$
and its distance from the origin is