Answer
A function $f(x,y)$ does not exist
Work Step by Step
Since, $f_{x}(x, y)=x+4y \\ f_{xy}(x, y)=4$ ...(1)
$ f_{y}(x, y)=3x-y \\ f_{yx}(x, y)=3$ ...(2)
As per Clairaut's Theorem we should have
$f_{xy}(x, y)=f_{yx}(x, y)$
Since, the second order derivatives are not equal in the both equations (1) and (2), thus, both $f_{xy}$ and $f_{yx}$ are continuous,
This means that such a function $f(x,y)$ does not exist.
