# Unique minimum of a simple function

• December 28th 2011, 07:07 AM
jens
Unique minimum of a simple function
Hi,

I want to prove that the following function has a unique minimum … and I'm surprised at how hard this is!

$f(x,y) = \frac{\frac{a}{x} + \frac{b}{y} + c}{d - e(x+y)} + \frac{u}{x} + \frac{v}{y}$

The scalars $a,b,c,d,e,u,v$ are all positive. The function is defined on the open triangle $x > 0$, $y > 0$, $x+y < \frac{d}{e}$.

Can you think of some simple proof?

I've tried a few things:
– take the logarithm
– try to show quasi-convexity (which is a stronger result) by several means
– try to rule out saddle points (see my post http://www.mathhelpforum.com/math-he...ts-194122.html and answer by xxp9)
– try to identify a composition of elementary functions
…to no avail

Below is a contour plot of the inverse $1/f$ (because it's nicer to plot than $f$). The function tends to zero on the border of the triangle. You clearly see there's a unique maximum of $1/f$ (minimum of $f$) somewhere inside the triangle.

Attachment 23144
• December 30th 2011, 02:21 AM
jens
Re: Unique minimum of a simple function
Hi again,

I figured out that my function $f$ is actually convex. One can show in a few steps that $f$ is a sum of functions that are all convex on the mentioned triangle.

For some reason (which I won't explain in detail) I had convinced myself that $f$ couldn't be convex. I failed to see the wood for the trees. You can consider this issue as closed (sorry for posting!).