Does the utility function U = x1^(0.5) * x2^(0.5) have strictly monotonic preferences?
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Consider the utility function $u\left(x_{1}, x_{2}\right)=\sqrt{x_{1} x_{2}}$. What kind of preferences does it represent? Is the function $v\left(x_{1}, x_{2}\right)=x_{1}^{2} x_{2}$ a monotonic transformation of $u\left(x_{1}, x_{2}\right) ?$ Is the function $w\left(x_{1}, x_{2}\right)=x_{1}^{2} x_{2}^{2}$ a monotonic transformation of $u\left(x_{1}, x_{2}\right) ?$
What kind of preferences are represented by a utility function of the form $u\left(x_{1}, x_{2}\right)=x_{1}+\sqrt{x_{2}} ?$ Is the utility function $v\left(x_{1}, x_{2}\right)=x_{1}^{2}+2 x_{1} \sqrt{x_{2}}+x_{2}$ a monotonic transformation of $u\left(x_{1}, x_{2}\right) ?$
Does the function have a global maximum? A global minimum? $$h(x, y)=1-y^{2} e^{x y}$$
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