Suppose we want to minimize the function: $\nu(x_1,x_2) = 9x_1^2 + 4x_2^2 + 12 \rho x_1x_2$, where $|\rho| \leq 1 $
on the set: $D:= \{2x_1+x_2 \geq \bar \pi, \ x \geq 0, \ x_1+x_2 = 1\}$, where $\bar \pi \in [1,2].$
My professor writes:
The minimum level curve crossing the region $D$ is an ellipsis tangent to $D$. If the tangency point $P$ lies in $D$ it will be the minimum point. Otherwise the minimum point will be one of the extreme of $D$ (the one closest to $P$).
If a level curve is tangent to $D$ at the point $P$, how is it possible that $P$ is not in $D$??