Find a function $z=f(x,y)$ whose partial derivatives are as given, or explain why this is impossible.
We have that $ f_x$ = $ 3x^2y^2-2x$, and $f_y$ = $ 2x^3y+6y$. where $ f_z$ denotes the partial derivative of the function $ f$ with respect to some variable $ z$.
I believe that given the partial derivatives, there is not a function $ z=f(x,y)$ whose partial derivatives are as given.
Pf: We will integrate both $f_x$ and $ f_y$ . The integral of $ f_x$ and the integral of $ f_y$ are not equal by calculus. QED.
Am I correct?