Example 1 Let $x, y \in \mathbf{R}$, find the minimum value of the function $f(x, y)=x^{2}+6 y^{2}-2 x y-14 x-6 y+72$, and find the values of $x, y$ when the minimum value is obtained.
From $f(x, y)=(x-y-7)^{2}+5(y-2)^{2}+3$, we know $f(x, y) \geqslant 3$. At this point, from $x-y-7=0$ and $y-2=0$, we get $x=9, y=2$ when $f(x, y)$ reaches its minimum value of 3.