I have a family of curves given by $g(x,y)=C_0 yx^{-n}$. How can I determine the function $f(x,y)$ for the family of curves that satisfies the condition that the local minima $\frac{\partial f}{\partial y}=0, \frac{\partial^2f}{\partial y^2}>0$ always falls on $g(x,y)$.
EDIT 1: The other derivative, i.e. $\frac{\partial f}{\partial x}$ at the point where $\frac{\partial f}{\partial y}=0$ is specified by means of an integral equation: $\int_0^H(\frac{\partial f}{\partial x} |_{\frac{\partial f}{\partial y}=0})dy=F_0$
There are many functions that have a local minimum on this curve. It would be more common to ask about functions of a given form with one or more parameters. Also your conditions for a local minimum only give a minimum along the $y$ direction. Usually you would require that $\frac {dF}{dx}=0$ and that all the second derivatives are greater than zero.