I encounter a function $$f(x,y)=-4xy \pi( \cos y)( \sin^2 (x\pi))+(\sin y)[-2y^2 x \pi +2x^3 \pi^3+(y^2+x^2 \pi^2)(\sin (2x\pi))] $$ defined for $x\times y\in[0,2]\times[0,\pi]$.
This function seems to be nonnegative in this compact domain. However, I just can prove this result after tedious analysis on the local property of $f$ around the zero points.
Since this function seems to be simple, I wonder whether there is a straightforward technique to do so .Any hints or references in this respect would be greatly appreciated!