How do I maximize the function $W=\min(u_1,u_2)$ subject to the constraint $u_1^2+u_2^2=100$?

Question

How do I maximize the function $W=\min(u_1,u_2)$ subject to the constraint $u_1^2+u_2^2=100$?

Usually I would do a Langrangian on the function to be maximized, but here it is difficult to do so.

The diagram I believe looks somewhat like this:

Answer 1

Prove first that one of the variables must be $\le \sqrt{50}$. Assume the contrary. Let $u_1, u_2 > \sqrt{50}$, then we have:

$$u_1^2 + u_2^2 > 50 + 50 > 100$$

A contradiction.

From this we conclude that maximum of W is $\le \sqrt{50}$. Now to find when the maximum of $W = \sqrt{50}$ occurs. Assume $u_1 \ge u_2 = \sqrt{50}$. Then we have:

$$100 = u_1^2 + u_2^2 \ge 50 + 50 = 100$$

From this we conclude that $u_1 = \sqrt{50}$. So the maximum of $W$ occurs when $u_1 = u_2 = \sqrt{50}$