# Find the critical points of a function f(x,y)=xy^{2}-3x^{2}-y^{2}-2x+2

Find the critical points of a function $f\left(x,y\right)=x{y}^{2}-3{x}^{2}-{y}^{2}-2x+2$
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Wendy Boykin
Partial derivatives of $z=f\left(x,y\right)=x{y}^{2}-3{x}^{2}-{y}^{2}-2x+2$ are $\frac{\partial z}{\partial x}={y}^{2}-6x+2$ and $\frac{\partial z}{\partial y}=2xy-2y=2y\left(x-1\right)$
Make them equal to zero to find the critical points: ${y}^{2}-6x+2=0,2y\left(x-1\right)=0$
The second equation will be true when $y=0$, then the first equation will be $-6x+2=0$ so that $6x=2$ and $x=\frac{1}{3}$. Now, we have one critical point: $\left(x,y\right)=\left(\frac{1}{3},0\right)$
The second equation wiil also be true when $x=1$. Then, the first equation will be ${y}^{2}-4=0$ and ${y}^{2}=4$, making $y=±2$. Now, we have two critical points: $\left(x,y\right)=\left(1,2\right)$ and $\left(x,y\right)=\left(1,-2\right)$
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Shannon Hodgkinson