Find Maxima and Minima Sir, help me to find the maxima, minima & saddle point.
$$f(x,y)=x^2-x^2y+2xy^2+4y$$
 here $$\begin{align}f_x(x,y) & =2x-2xy+2y^2
\\      f_{xx}(x,y) & =2-2y
 \\     f_y(x,y) & =-x^2+4xy+4
  \\    f_{yy}(x,y) & =4x
   \\   f_{xy}(x,y) & =-2x+4y
\end{align}$$
For critical point I put $f_x=0$ & $f_y=0$, but fail to subsitute $y=(x^2-4)/4x$ to find the value of $x$ in the equation $-x^2+4xy+4=0$

setting $$f_x = 0  \to x = \frac{y^2}{y-1} \, or\, y = 1 $$ in the case $y = 1, x = \pm 2\sqrt 2$ and in the other case we have $$0 = f_y \to 0 = -\frac{y^4}{(y-1)^2} + \frac{4y^3}{y-1} + 4 \to -y^4+4y^3(y-1)+4(y-1)^2=3y^4-4y^3+4(y-1)^2 $$ the last equation has two roots $$y = 0.634, 1.2828 , x = \frac{y^2}{y-1} \text{ and }x = \pm 2\sqrt 2, y = \pm 1.$$
you can take it from here.
