want a detail solution not just answer

[N]

a firm uses 4 inputs to produce 1 output. the production function is f(x, y, u, v) = min(x, y)+min(u,v)
(a) what is the vector of conditional factor demands to produce 1 unit of output when the factor price vector is w = (1, 2, 3, 4)?

another firm has a production function f(x,y,u,v) = min(x+y, u+v). what is the vector of conditional factor demands to produce 1 unit of output when prices are w=(1,2,3,4)?
what is the cost function of the firm?

[Dreyfus]

For the first firm,
The production technology requires either x=y as input or u=v or the combination of both (depending on factor prices!
Given the factor prices w = (1, 2, 3, 4) consider:
Case1 when firm only uses x&y for producing output.
Since firm has to employ 1 unit of each x&y to produce 1 unit of output the total cost of producing 1 output is (1+2)=3
Case2when firm only uses u&v for producing output.
Since firm has to employ 1 unit of each u&v to produce 1 unit of output the total cost of producing 1 output is (3+4)=7.
Clearly cost is minimum when x&y are used.
And in general the conditional factor demands are:
(a,a,0,0) if wx+wy<wu+wv
(0,0,a,a) if wx+wy>wu+wv
(a,a,1-a,1-a) if wx+wy=wu+wv
For the second firm, the production function requires that either x or y(or both) must be used with either u or v(or both) to produce output.
So conditional input demands in this case is
(a,0,a,0) if wx<wy and wu<wv
(a,0,0,a)    wx<wy and wu>wv
(a,0,a,1-a) wx<wy and wu=wv
Similarly u can proceed with other cases
So given the factor prices, the cost of producing 1unit of output is (1+3) =4