Example 4 Let $x, y \in \mathbf{R}$, find the minimum value of the function $f(x, y)=x^{2}+$ $6 y^{2}-2 x y-14 x-6 y+72$.


Solution: $f(x, y)=(x-y-7)^{2}+5(y-2)^{2}+3$, so when $x=9, y=2$, $f_{\text {min }}=3$.