Problem: Determine whether or not the following optimization problem has a solution: $f(x,y,z) = x^{sin(10)} + e^{x+y} + x^3y^{201} + sin(20z)$, subject to: $h(x,y,z) = x^6 - (1/3)x^3y + y^2 + z^4 \le 2011$.
Hint: make use of $|mn| \le a^{-1}|m|^a + b^{-1}|n|^b$, for $a^{-1} + b^{-1} = 1$.
Attempt at solution: There seem to be some random numbers in the setup of the problem which led me to believe that there should be some simple solution to it, however I am unable to find one. By inspection the problem doesn't seem to exclude the possibility of having an answer, since $f$ is continuous and $h^{-1}$ seems to be continuous (I'm not sure about this) so $f$ should take on a minimum on the domain... however this approach makes no use of the hint provided which makes me suspicious if $h^{-1}$ is actually continuous. The other approach I had was using Kuhn-Tucker, but it led nowhere since the gradient of $f$ and $h$ are very messy. Can someone suggest a better way to approach this problem? Or am I missing something obvious? Thanks.