How can I determine whether or not a function exists given two partial derivatives?

Question

"Can there exist a $C^2$ function $f(x,y)$ with $f_x = 2x-5y$ and $f_y=4x+y$"?

Given this question, am I simply to take the second derivative of these functions to prove the equivalence of the mixed partial derivative?

Answer 1

With $f_x = 2x-5y$ and $f_y= 4x+y$, there is no $C^{2}$ function for which these relations hold.

Simply check, $f_{xy} = -5$ and $f_{yx}=4$ which are different.

Answer 2

Given that in the mixed partial derivative the order in which you take the derivatives is not important, you could try do compute $f_{xy}=\frac{f_x}{\partial y}$ and $f_{yx}=\frac{f_y}{\partial y}$ and compare them. If they are indeed equal then $f_x$ and $f_y$ may come from the same primitive.