<s>[INST] Find positive values such that $xy = 32$ and the sum $4x+y$ is as small as possible. 
Find positive values such that $xy = 32$ and the sum $4x+y$ is as small as possible.

How can I solve this? I know that the answer is $2\sqrt{2} = x$ and $8\sqrt{2} = y$, but I can't seem to figure out a concrete method for solving this type of problem.
An explanation of how to solve the question would be greatly appreciated!
 [/INST] We can write: $4x+y = \left(2\sqrt{x}-\sqrt{y}\right)^2+ 4\sqrt{xy}= \left(2\sqrt{x}-\sqrt{y}\right)^2+ 16\sqrt{2}\geq 16\sqrt{2}\implies \text{min} = 16\sqrt{2},$ it occurs when $\sqrt{y}=2\sqrt{x}\implies x = 2\sqrt{2}, y = 8\sqrt{2}$.
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