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Question Number 96185 by bemath last updated on 30/May/20

$$\mathrm{what}\mathrm{are}\mathrm{critical}\mathrm{points}\mathrm{of}\mathrm{this}$$$$\mathrm{function}\mathrm{z}=\mathrm{xy}+{5\mathrm{xy}}^2+10\mathrm{y}$$

Answered by john santu last updated on 30/May/20

$$\left(1\right)\frac{\partial \mathrm{z}}{\partial x}=y+5y^2=0$$$$y(1+5y)=0\{\begin{array}{l}y=0\\ y=-\frac{1}{5}\end{array}$$$$\left(2\right)\frac{\partial \mathrm{z}}{\partial \mathrm{y}}=x+10x\mathrm{y}+10=0$$$$\Rightarrow x(1+10\mathrm{y})=-10.$$$$\mathrm{if}\mathrm{y}=0\Rightarrow x=-10$$$$\mathrm{if}\mathrm{y}=-\frac{1}{5}\Rightarrow x=10$$$$criticalpointare(-10,0);$$$$(10,-\frac{1}{5}).$$$$$$

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