APEX Calculus, p.711, Ex. 29

In Exercises 27 – 30, form a function \(z = f(x, y)\) such that \(f_{x}\) and $f_{y} match those given.

  1. \(f_{x}=6xy-4y^{2}\), \(f_{y}=3x^{2}-8xy+2\)

\[\int f_{x}dx=\]

\[=\int 6xy-4y^{2}dx\]

\[=3x^{2}y-4xy^{2}+C\]

\[\int f_{y}dy=\]

\[\int 3x^{2}-8xy+2dy=\]

\[3x^{2}y-4xy^{2}+2y+C\]

Adding all of the unique terms we get:

\[f(x,y) = 3x^{2}y-4xy^{2}+2y+C\]