I'm looking at a problem where the following two equations are given: $$ xy \ge C $$ and $$ a \le {\frac{x}{y}} \le b, $$ where $x> 0, y >0$.
Then they say at which values min/max values for $x$ and $y$ will be got, namely: $$x_{min}=\sqrt {Ca}, x_{max}=\sqrt {Cb}, y_{min}=\sqrt {{\frac{C}{b}}}, y_{max}=\sqrt {{\frac{C}{a}}}$$
I'm trying to understand how this is implied from the two equations. So, from he first equation one can say that $x\ge {\frac{C}{y}}$. By plugging that in the second equation we get $a \le {\frac{C}{y^2}} \le b$. Here, I'm not sure that all equations should be kept as in the original equation. If being able to keep it, I'd get $ay^2\le C$, from where I'd derive min and max values for $y$.
So can someone explain how exactly to derive the four variables?