Thread: Maxima & Minima of Multivariable Function

1. Maxima & Minima of Multivariable Function

Asked to find all maximum, minimum and saddle points to the function $\displaystyle f(x)=x^2+xy+y^2+\frac{1}{x}+\frac{1}{y}$. x and y both > 0.

The condition that x and y are both positive allows us to multiply through variables without fear of multiplying by 0.

After computing the first derivatives I am having a very hard time finding the critical points by solving the result system of equations, and was trying to see if anyone could help out. I don't need anything else beyond finding the critical points (the rest is just grunt work really).

Here's the system of equation after computing the first partial derivatives and setting them to 0. I've tried a dozen times to solve it, and I keep getting garbage!

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

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

Thanks for anyone that helps out!

2. I cheated and used wolframalpha.com to get the answer.

$\displaystyle x=\frac{1}{3^\frac{1}{3}}$

$\displaystyle y=\frac{1}{3^\frac{1}{3}}$

It didn't give steps for solving though. But maybe now that you know the answer you'll have a better idea of how to get to it. Just to clarify, that's 1 divided by the cubed root of 3 for both x and y.

3. Look at the symmetry. Look for solutions such that y= x.