Question

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

Answer 1

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 \(f_y=8xy-2x^2+3x\).
\[f_x=0 \implies 4y^2-4xy+3y=0 \implies y(4y-4x+3)=0\]
\(\large \implies y=0 \text{ or } y=x-\frac{3}{4}.\)

Answer 2

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 \((0,0), (\frac{3}{2},0), (0,-\frac{3}{4}), (\frac{1}{2},-\frac{1}{4}). \)