Standard optimization problem using Lagrange Multipliers

Hello Everyone!

I'm a bit stuck on this one, and was wondering whether you could give me a hint to go about it:

.

Now we can move the constraint and solve this new problem:

.

Taking gradients:

Now, I know I need to find and in terms of , something that would yield one variable in the terms of another and which made me throw that whole approach and think about these two problem:

, and

Both problems would yield that .

I would find the minimum in each case, then rule out the greater of these two minimums. Would this work?

Any help is appreciated!