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

Monique Slaughter
2021-12-16
Answered

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

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Wendy Boykin

Answered 2021-12-17
Author has **35** answers

Partial derivatives of $z=f(x,y)=x{y}^{2}-3{x}^{2}-{y}^{2}-2x+2$ are $\frac{\partial z}{\partial x}={y}^{2}-6x+2$ and $\frac{\partial z}{\partial y}=2xy-2y=2y(x-1)$

Make them equal to zero to find the critical points:${y}^{2}-6x+2=0,2y(x-1)=0$

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: $(x,y)=(\frac{1}{3},0)$

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 $y=\pm 2$ . Now, we have two critical points: $(x,y)=(1,2)$ and $(x,y)=(1,-2)$

Make them equal to zero to find the critical points:

The second equation will be true when

The second equation wiil also be true when

Shannon Hodgkinson

Answered 2021-12-18
Author has **34** answers

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