Example 6. Determine the range of the curve
$$
2 x^{2}-2 x y+y^{2}-6 x-4 y+27=0
$$

in the coordinate plane.
Solve $A=B^{2}-4 A C=-i<0$, this curve represents an ellipse.
The derivatives are $f_{x}^{\prime}(x, y)=4 x-2 y-6=0$,
$$
f_{y}^{\prime}(x, y)=-2 x+2 y-4=0
$$

Substituting into the curve equation $(5 \cdot 2)$, we find that the intersection points of the line $f_{x}^{\prime}(x, y)=0$ and the curve $(5 \cdot 2)$ are $P_{1}(4,5)$, $P_{2}(6,9)$; the intersection points of the line $f_{y}^{\prime}(x, y)=0$ and the curve $(5 \cdot 2)$ are $P_{1}^{\prime}(2+\sqrt{5}, 4+\sqrt{5})$, $P_{2}^{\prime}(2-\sqrt{5}, 4-\sqrt{5})$.

Therefore, by Corollary 1 and Corollary 2, the curve (5.2) lies within the rectangular region bounded by the four lines $y=5$, $y=9$, $x=2+\sqrt{5}$, and $x=2-\sqrt{5}$.