Consider the function $f(x,y) = x(1-y)\log(1+y/x^2)$, where $0\le x, y\le1$. Is this function quasi-concave?

  • $\begingroup$ Do you have any reason to believe it is? I know some quasiconvex functions are quite nonintuitive but I would have no reason to expect this one to be. $\endgroup$ – Michael Grant Mar 25 '16 at 19:22
  • $\begingroup$ I believe this function is quasi-concave due to the following reasons: 1) The second derivative with respect to y is $-x(2x^2+y+1)/(x^2+y)^2$ so that $f(x,y)$ is concave with a fixed $x\ge 0 $. 2) when I plot the graph $f(x,y)$ it looks like a quasi-concave function. However, I don't know how to prove mathematically. $\endgroup$ – KJ Choi Mar 25 '16 at 19:42