How do I solve the following problem

Question

Let's suppose I have the following function $f$:

$$f: \mathbb{R}^2\setminus (0,0) \to \mathbb{R}$$ $$(x,y) \mapsto f(x,y) = \frac{1}{x} - \frac{1}{2y} - \frac{y}{2}$$

If I want to find the maxima and minima, the partial derivatives need to be calculated and be equal to zero. Therefore,

$$\frac{\partial{f}}{\partial{x}}= \frac{\partial{f}}{\partial{y}} = 0$$ $$ x^{-2} = -\frac{y^{-2}}{2}+\frac{1}{2} = 0 $$

So, what should I do now? Consider the following as maxima and minima? Should I call them as supremum and infimum? Should I change $\mathbb{R}$ to $\mathbb{\bar{R}}$?

Values:

$$\{(-\infty,-1),(-\infty,1),(\infty,-1),(\infty,1)\}$$

Obs.: If so, they are all maxima or minima according to the 2nd derivative test.

Thanks

Answer 1

The simplest method to observe that the given function is unbounded above and below is to note that $$f\left(\dfrac1{n+1},1\right)=n\quad\forall n\in\Bbb N\\ f\left(4n,2n\right)=-n\quad\forall n\in\Bbb N$$

Answer 2

Note that there is no $x$ such that $x^{-2}=0$. Therefore there are no local maxima or minima. Writing the single line with two equals signs may conceal this.