Inequality problem

Question

1. $5\geq X \geq 2$

2. $0.8 \geq Y\geq 0.5$

Find range of the values of

A) $\sqrt{XY}$

B) $(\frac{X}{Y})^2$

Answer 1

$\sqrt{XY}$ is increasing in both $X$ and $Y$. It can be as small as $\sqrt{2*0.5}$ and as large as $\sqrt{5*0.8}$

$(\frac{X}{Y})^2$ increases in $X$ but decreases in $Y$ it can be as large as $(\frac{5}{0.5})^2$ and as small as $(\frac{2}{0.8})^2$

Answer 2

Intuitively the largest value of $xy = 5 \times 0.8$ while the smallest value is $2 \times 0.5$. So you have your range of xy. I hope you can find the range of $\sqrt{xy}$ from there. Also intuitively for $\dfrac{x}{y}$ the highest value is $\dfrac{5}{0.5}$ and the lowest value is $\dfrac{2}{0.8}$. From there you can find the range of their squares.