Find the local maximum, minimum and saddle points of $f(x,y) = x + y + x^2y + xy^2$

Question

Find the local maximum, minimum and saddle points of $f(x,y) = x + y + x^2y + xy^2$

$f_{xy} $ is not equal to $f_{yx} $, therefore nothing can be said about the critical points of this function.

Is this correct?

Answer 1

We have:

$$\frac{\partial f}{\partial x}=1+2xy+y^2 \qquad \frac{\partial f}{\partial y}=1+2xy+x^2 $$ so $$ \frac{\partial^2 f}{\partial x\,\partial y }=2x +2y \qquad \frac{\partial^2 f}{\partial y\,\partial x}=2x +2y $$

so the mixed second derivative are equal and the problem does not exists.

Answer 2

solve the equationsystem $$1+2xy+y^2=0$$ $$1+x^2+2xy=0$$ for your function is hold $$-\infty<f(x,y)<\infty$$