Math Help - Reverse engineering cost functions

Hi, I have a problem where a cost function is given with attributes y: output, w: the cost of labour l and r: the cost of capital k .

THe question involves to find the corresponding production function from a given cost function. I can do the cost minimization problem when one says that :

f(k,l)=y (constraint) and C(k,l)=wl+rk

To find the cost function we simply need to use lagrange multipliers, find the cost minimizing capital and labour, and insert them in the cost function, which then becomes the minimizing cost function:
c(y,w,r)= ......

I can't figure out how to do the opposite though.

Statring from
c1(y,w,r,) = (y/2)*(w+r)
and
c2(y,w,r)= 2y*sqrt(wr)

can you explain to me how to find the corresponding production functions f1 and f2?

Thanks


EDIT:

I found the two functions. Correct me if i'm wrong:

f1(l,k)= 2*min{l,k}
f2(l,k)=l^1/2 * k^1/2

my approach was however trial and error with a bit of intuition. Does anyone have a systematic way of finding them?

i seriously doubt (although i haven't proved) that there is a unique production function for a given cost function.

intuition: ignoring corner solutions to the cost minimisation problem, the minimum cost function is the set of poins where isoquant curve has the same slope as the isocost curve. It doesn't contain a full description of any other points of the production function, all you know is that they dont satisfy the slope condition above.


Edit: in fact there is a simple counter example: consider a linear production function where the slope is great enough to induce a corner solution, increasing the slope further will retain the corner solution and hence the minimum cost function. Since more than one production function gives the same cost function, no algorithm can recover the production function from the cost function in the general case.