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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.

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  • $\begingroup$ What went wrong with looking at the second derivative? $\endgroup$ – littleO Sep 15 '16 at 8:54
  • $\begingroup$ 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. $\endgroup$ – C. Klum Sep 15 '16 at 9:20
  • $\begingroup$ I haven't worked out the details, but it seems like you might have been on the right track. $\endgroup$ – littleO Sep 15 '16 at 9:21

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