Maximum or minimum in a function I Have a function that doesn't exit on (0,0). I want to apply second derived criteria, to this function, to know if that point is maximum or minimum, but the second derived doesn't exits on (0,0) either. I know is a critical point but is not either a maximum or a minimum. So what is it? The function is $$f(x, y) = xy + \frac1x + \frac1y$$
 A: Given:
$$f(x,y)=xy+\frac{1}{x}+\frac{1}{y} \tag1$$
$$\frac{\partial \:}{\partial \:x}\left(xy+\frac{1}{x}+\frac{1}{y}\right)=y-\frac{1}{x^2}=0$$ 
Gives: $$y=\frac{1}{x^2} \tag2$$
$$\frac{\partial \:}{\partial \:y}\left(xy+\frac{1}{x}+\frac{1}{y}\right)=x-\frac{1}{y^2}=0$$
Gives $$x=\frac{1}{y^2}$$
Which leads to $y^2=\frac{1}{x}$
Using (2) with the above result we have the system:
$$y=\frac{1}{x^2}$$
$$y^2=\frac{1}{x}$$
This system has a single real solution $x=1$. Which implies, $y=1$
Now we have 1 critical point (1,1).
To determine whether this point is max. or min. you take the 2nd partial derivative w.r.t $x,y$ and test it using (1,1)
$$\frac{\partial \:}{\partial \:x}\left(y-\frac{1}{x^2}\right)=\frac{2}{x^3}$$
$$\frac{\partial \:}{\partial \:y}\left(x-\frac{1}{y^2}\right)=\frac{2}{y^3}$$
We now need to determine the type of the extreme using the following test where (a,b) is the extreme point, that is (1,1) in this case:
$$v=\frac{\partial ^2}{\partial \:x^2}f\left(a,b\right).\frac{\partial \:^2}{\partial \:\:y^2}f\left(a,b\right)-\left[f\left(a,b\right)\right]^2$$
$$v(1,1)=2*2-9=-5$$
Since $v(1,1)<0$, all we can say is that the point (1,1) is a Saddle Point.
A: Unless you extend the definition of your function to give it some value at the point $\ (0,0)\ $, that point is not a critical point of the function according to any of the commonly used definitions of that term, most of which require a critical point of a function to lie within the domain of definition of that function.
The expression $\ \displaystyle xy + \frac{1}{x} + \frac{1}{y}\ $ is not defined whenever $\ x=0\ $ or $\ y=0\ $, and can have arbitrarily large or arbitrarily small values at points $\ (x,y)\ $ arbitrarily close to $\ (0,0)\ $, so no matter how you extend its definition to give it a value at $\ (0,0)\ $, that point cannot be either a maximum or a minimum. A point with these characteristics is called a singularity.
