# Domain of a function, while keeping it convex

Let $f(\phi)=\phi^\alpha \phi^\beta -\phi^\alpha$, be a function where $\alpha,\beta\in \mathbb{N}$. Of which domain is the function convex?

I have tried using the Definition for convexity: $(1-\lambda)f(x)+f(y)\leq f((1-\lambda)x+\lambda y)$ and using the method of double differentiation, where $0\leq f''(x)$

But I havent made any real progress, would appreciate some help.

• What went wrong with looking at the second derivative? – littleO Sep 15 '16 at 8:54
• I had to divide with $x^\alpha$, which then changes the inequality, depending on being negative or not. So then I would get a couple of different intervals. Which depends on $\alpha$ being even, or being uneven while x is negative. And that didnt seem to make sense for me. – C. Klum Sep 15 '16 at 9:20
• I haven't worked out the details, but it seems like you might have been on the right track. – littleO Sep 15 '16 at 9:21