## anonymous 4 years ago locate and classify all critical points for f(x,y)=4ypower2 x - 2yxpower2 + 3xy

1. Mr.Math

We want to find the critical points of $$f(x,y)=4y^2x-2yx^2+3xy$$. The critical points of such a polynomial is the points satisfying $$f_x=0,f_y=0$$ simultaneously. So lets first find the partial derivatives: $$f_x=4y^2-4xy+3y$$ and $$f_y=8xy-2x^2+3x$$. $f_x=0 \implies 4y^2-4xy+3y=0 \implies y(4y-4x+3)=0$ $$\large \implies y=0 \text{ or } y=x-\frac{3}{4}.$$

2. Mr.Math

Substituting $$y=0$$ into $$f_y=0$$ gives us: $f_y|_{y=0}=-2x^2+3x=0 \implies -x(2x-3)=0 \implies x=0 \text{ or } x=\frac{3}{2}.$ Substituting $$y=x-\frac{3}{4}$$ gives: $\small 8x^2-6x-2x^2+3x=6x^2-3x=3x(2x-1)=0 \implies x=0 \text{ or } x=\frac{1}{2}.$ $$x=0 \implies y=-\frac{3}{4} \text{ and } x=\frac{1}{2} \implies y=-\frac{1}{4}$$. Thus the critical points are $$(0,0), (\frac{3}{2},0), (0,-\frac{3}{4}), (\frac{1}{2},-\frac{1}{4}).$$