A firm has determined that its cost of service C in thousand

A firm has determined that its cost of service C, in thousands of dollars, is given by C(x) = 3x+8+12/x-7, 8 lessthanorequalto x lessthanorequalto 15, Where x represents the number of \"quality units\". Find the number \"quality units\" that the firm should use in order to minimize the cost of service. note that minimum cost of service here must be the absolute minimum of C on the interval [8,15]

Solution

C(x) = 3x + 8 + 12/(x - 7) ; 8 <= x <= 15

C \'(x) = 3(1) + 0 - 12/(x - 7)² ; since d/dx xⁿ = n x^n-1 , d/dx 1/x = -1/x²

==> C \'(x) = 3 - 12/(x - 7)²

critical points ==> C \'(x) = 0

==> 3 - 12/(x - 7)² = 0

==> 12/(x - 7)² = 3

==> (x - 7)² = 4

==> x - 7 = 2 or x - 7 = -2

==> x = 2 + 7 or x = -2 + 7

==> x = 9 , x = 5

==> x = 9 ( x cannot be 5 since 8 <= x <= 15)

C(9) = 3(9) + 8 + 12/(9 - 7)

==> C(9) = 27 + 8 + 6 = 41

checking end points i.e, C(8) , C(15)

==> C(8) = 3(8) + 8 + 12/(8 - 7) = 24 + 8 + 12

==> C(8) = 44

C(15) = 3(15) + 8 + 12/(15 - 7) = 45 + 8 + (3/2)

==> C(15) = 109/2

Hence minimum at x = 9 and minimum value = C(9) = 41