A firm with the production function f (x1, x2, x3, x4)  minx1, x2, x3, x4 faces input prices w1  $1, w2  $2,w3  $2, w4  $5 for factors 1, 2, 3, and 4.The firm must use at least 17 units of factor 2.The lowest cost at which it can produce 100 units of output is

A) $1,000.
B) $400.
C) $317.
D) $615.
E) $300.

To minimize cost, the firm should use the input combination that equates the marginal product of each input to its price.

Let the marginal products (MPs) of inputs 1, 3, and 4 be denoted by MP1, MP3, and MP4, respectively. Then, the marginal product of input 2 is:

MP2 = 4x2/x1 = 40/x2

The cost-minimizing input combination is:

MP1/w1 = MP2/w2 = MP3/w3 = MP4/w4

Solving for x1, x3, and x4, we get:

x1 = (10/3)x2
x3 = (5/2)x2
x4 = x2

The cost of producing F1F1F1100 units of output is:

C = w1x1 + w2x2 + w3x3 + w4x4
C = $1(10/3)x2 + $2x2 + $2(5/2)x2 + $5x2
C = $31.7x2

However, the firm must use at least 17 units of factor 2. Thus, the minimum cost of producing 100 units of output is:

C = $31.7(17) = $538.9

Therefore, the closest option to the minimum cost is Option C ($317).