Question Number 203859 by Mastermind last updated on 30/Jan/24
$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{surface}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{z}\:=\:\mathrm{xy} \\ $$$$\mathrm{has}\:\mathrm{neither}\:\mathrm{a}\:\mathrm{maximum}\:\mathrm{nor}\:\mathrm{a}\:\mathrm{minimum}\:\mathrm{point} \\ $$
Commented by AST last updated on 30/Jan/24
$${For}\:{x},{y}\rightarrow\infty;\:{z}\rightarrow\infty \\ $$$${For}\:{x}\rightarrow\infty;\:{y}\rightarrow−\infty\Rightarrow{z}\rightarrow−\infty \\ $$
Commented by Mastermind last updated on 30/Jan/24
$$\mathrm{I}\:\mathrm{need}\:\mathrm{full}\:\mathrm{detail},\:\mathrm{using}\:\mathrm{pdf}\:\mathrm{test} \\ $$
Commented by AST last updated on 30/Jan/24
$$\frac{\partial{z}}{\partial{y}}={x};\frac{\partial{z}}{\partial{x}}={y}…\:{Equating}\:{to}\:\mathrm{0}\Rightarrow\left({x},{y}\right)=\left(\mathrm{0},\mathrm{0}\right) \\ $$$${You}\:{also}\:{have}\:{to}\:{check}\:{boundary}\:{points},\:{which} \\ $$$${is}\:{what}\:{is}\:{up}\:{there}.\:{Partial}\:{differentials}\:{do}\:{not} \\ $$$${give}\:{all}\:{critical}\:{points}. \\ $$