Find critical points of the following function $f(x,y)$

Question

I tried to find the critical points of the function

$$f(x,y) = x^2y-2xy + \arctan y $$

And I found that is $P(1,0)$, the problem is that the Hessian is null, and I don't know how to procede to determine the nature of that point. Can you help me ?

Update: Thanks you all, and I tried to study the sign of the function, the problem is that I don't know how to proceed , since I have $Δf(x,y)=x^2y-2xy + \arctan y $ and I don't know how to study the sign locally around $1,0$.

Answer 1

$f(1,0)=0$, $f(1,1)<0$ and $f(3,1)>0$, so $(1,0)$ is a saddle point.

Answer 2

for your function $f(x,y)$ holds $$-\infty<f(x,y)<+\infty$$

Answer 3

Generally in cases like this you just have to get your hands dirty. We have that f(1,0) = 0. You want to look at what happens when you vary x and y around (1,0), in various ways. If you can convince yourself that the result is universally positive for (x,y) close enough to (1,0) then this point is a minimum, if it's negative then (1,0) is a maximum, and if you can find, arbitrarily close to (1,0), both points that yield positive values and points that yield negative values then (1,0) is a saddle point.

(Of course, the process by which you do this should be informed and thoughtful, not random. You need to carefully consider the structure of the function in question).