Proof that geometric mean of two variables with range from $0$ to $1$ maximizes when both approach 1.

Question

I am looking for a formal proof, that geometric mean of two variables that are between $0$ and $1$ is maximized as both approach $1$.

$G_m=\sqrt{x\times y}$

It is easy to intuitively see that $G_m$ would increase if both $x$ and $y$ increase at the same time. But I am having trouble to express it in a concrete mathematical fashion.

Answer 1

You are trying to solve $\max_{x\in[0,1],y\in[0,1]}\sqrt{xy}$? If so, note that $xy\leq1$ for any $(x,y)\in[0,1]^{2}$ so that $(x,y)=(1,1)$ is the solution to the problem. To see that it is the unique solution, note that $xy<1$ for any $(x,y)\in[0,1]^{2}$ not equal to $(1,1)$.