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Find $f$$(x,y)$ given $f_x$$= $$3x^2y-4y^2$ and $f_y$$=$$x^3-8xy+6y$ and $f$$(1,1)$$=$$5$

Can anyone steer me in the right direction here? I assume I have to integrate but we haven't done integration with multivariable functions yet. Any help at all is appreciated.

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Integrate with respect to $x$ and $y$ your partial derivatives. In both cases, you'll get a constant term which in this case is a function of y or x respectively.

For example $$\int f_x dx = x^3y-4xy^2+c(y)$$

Do the same for $f_y$ and set them equal and solve.

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    $\begingroup$ Alternatively, differentiate $x^3 y - 4xy^2 + c(y)$ partially with respect to $y$ (not forgetting the the $c'(y)$ term), set equal to $x^3 - 8xy + 6y$, and solve for $c'(y)$. $\endgroup$ – Theo Bendit Oct 22 '18 at 0:21

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