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i'm having some problem in establishing the convexity/concavity of the following two functions. Check for the concavity/convexity of the following functions: (a) $f_1:\mathbb{R}^2_+\rightarrow\mathbb{R}$ defined as $f_1(x, y) =\min\{\max\{x, 2y\}, \max\{2x, y\}\}$ (b) $f_2:\mathbb{R}^2_+\rightarrow\mathbb{R}$ defined as $f_2(x, y) =x^2 + y^2$. thank you.

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    $\begingroup$ For $f_2$ you may want to calculate its Hessian matrix and check that it's definite positive. If you're not into that stuff, you can use the fact that it can be written as $f_2(x,y) = g(x)+h(y)$ where $g$ and $h$ are two convex functions on $\mathbb R$ and use the definition (with $\lambda U + (1-\lambda)V$ ...) to prove it. $\endgroup$ – Alexandre Halm Oct 15 '14 at 7:51
  • $\begingroup$ If my answer for $f_1$ is OK for you, please just accept it. $\endgroup$ – Alexandre Halm Oct 15 '14 at 13:24
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In addition to the other answer and my other comments, for (a) $f_1$ is neither convex nor concave.

To see (and prove) it, just consider the intersection of the surface $z=f_1(x,y)$ with the vertical plane $x+y=10$. It's this 1D curve (parametrized by $x$) : $x \mapsto f_1(x,10-x)$.

You can see a plot here (hopefully). Clearly this is neither convex nor concave and provides sufficient counter-examples points to prove it.

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