Consider the following problem: Minimise $x_1^2 + 2x_2^2$ subject to $(x_1)^2 + x_2^2 \le 1$ and $x_2 = 1$.
Sketch the feasible set and the level sets of the objective function, and determine an optimal point $x*$ and the corresponding optimal value $p*$.
For the feasible set I know there will be a line parallel to the $x_1$ axis passing through $(0,1)$ and for the second condition there will a circle of unit radius centred at $(1,0)$ but I am unsure what the level sets are.
Please can someone let me know what a level set is and how to sketch it. I've been told it will be an oval but I don't understand why.
Level sets of function $z=x_1^2+2x_2^2$ are the curves on which this surface assumes the same value $c$. In other words, they are the curves $x_1^2+2x_2^2=C$ on the $x_1x_2$-plane. So for example, sketch the ellipses
$$x_1^2+2x_2^2=1\\ x_1^2+2x_2^2=2\\ x_1^2+2x_2^2=3\\ x_1^2+2x_2^2=4$$
gives you