Find the critical points of this function:

[knowlewj01]

Homework Statement

Find the critical points of this function and determine wether they are local maxima, minima or saddle points...

$f=\frac{1}{x} + \frac{1}{y} + xy$

The Attempt at a Solution

start off by partially differentiating and setting to zero for x and y:

$\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=0$


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

$\therefore y=\frac{1}{x^2}$


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

$\therefore x=\frac{1}{y^2}$

so:

$\exists$ a stationary point at $(\frac{1}{y^2},\frac{1}{x^2})$

using $D = f_{xx}f_{yy} - (f_{xy})^2$




Does all of this look ok so far? because i do not get a definitive answer for where the critical point lies or any indication of it's nature.

 

[Fightfish]

No, both conditions have to be satisfied. Surely you can't leave your coordinates in terms of each other - they can and should then be simplified! So, it is a matter of solving the equations
$$y=\frac{1}{x^2}$$
$$x=\frac{1}{y^2}$$
simultaneously.