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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!

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

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

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

Attachment 23144

Re: Unique minimum of a simple function

Hi again,

I figured out that my function $\displaystyle f$ is actually convex. One can show in a few steps that $\displaystyle f$ 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 $\displaystyle f$ couldn't be convex. I failed to see the wood for the trees. You can consider this issue as closed (sorry for posting!).