# How to verify the following function is convex or not?

Consider function $$f(x)=\frac{x^{n_{1}}}{1-x}+\frac{(1-x)^{n_{2}}}{x},x\in(0,1)$$ where $n_{1}$ and $n_2$ are some fixed positive integers.

My question: Is $f(x)$ convex for any fixed $n_1$ and $n_2$?

The second derivation of function $f$ is very complex, so I wish there exists other method to verify convex property.

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Yeah, you can always use the definition ;-) –  dtldarek Apr 22 '12 at 12:51

A method to show f is convex is to show $f''(x)>0$. Do the two terms separately, reduce to determining the sign of a quadratic polynomial in the numerator.
A real valued function f : X → R defined on a convex set X in a vector space is called convex if, for any two points x1,x2 in X and any t belongs [0 1] we have $f(t*x1+(1-t)*x2)<=(t*f(x1)+(1-t)f(x2))$ now let's take $n1$ and $n2$ some fixed values,let say 5 and 10,and try it