Prove convexity of complicated rational function Can anyone help me prove the convexity of this rational function? The man who proved the convexity of function used these facts. But I don't know this fact is correct or not.
Here are the facts and function: 


*

*As N increases, f(N) goes to infinity, That implies that there must be a minima (either at N=0 or somewhere else with a finite N)

*There cannot be more than one (positive) minima since we're dealing with second order equation.


f(N) =    +-cN^4+-dN^3+-eN^2+-fN+-g / +-aN^2+-bN  
a,b,c,d,e,f,g is constants. and N >=1. I guess the second order equation means that between the leading coefficient of the numerator and the denominator is 2. 
Are these facts correct? I think fact 1 is no problem, but fact 2 is correct or not.
I am waiting for any answers. thank you.

Fact $2$ seems incorrect (but I may not be understanding your notation). Local minima correspond to points where the derivative vanishes. For a rational function of the form $p/q$ where $p$ is of degree $n$ and $q$ is of degree $m$, the numerator of the derivative is a polynomial of degree $n+m-1$ and as such may have up to $n+m-1$ roots, which lead to at most $n+m-1$ possible local minima. In this case, there may be as many as $5$ local minima.
