I want to show that a function $f$ in three variables $(x,y,z)$ is non-negative on a specified subdomain: $0<=x,y,z<=1$.
The function comprises the following terms: $1,x,x^2,y,y^2,z,z^m,z^{m+1}$ where $m$ is a positive integer. In $f$, the terms appear in all combinations e.g. $1x^2$ or $x^2y^2z^{m+1}$.
My strategy to show this is as follows:
- Compute roots of $f$ w.r.t. $x$ --> $x_1, x_2$, i.e., two functions of $y,z$.
- Investigate limit of $f$ if $x$ goes to $-\inf$
My results are as follows. Solving for the roots of $f$ w.r.t. $x$ (using the formula for polynomials of degree 2) yields two functions of $x,y$ that describe the "x-position" of the roots.
In my particular case, these roots turn out to lie in the following ranges: $x_1\in(-\inf,0]$ and $x_2\in[1, \inf)$.
The limit of $f$ if $x\rightarrow -\inf$ is found to be $-\inf$.
Can I conclude from these findings that $f$ is non-negative on the specified domain?