checking for convexity/concavity of a function

Question

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.

Answer 1

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.