Question

In a firm the cost function for output x is given as C = `"x"^3/3 - 20"x"^2 + 70 "x"`.  Find the 3 output for which marginal cost  (C_m) is minimum.

Answer 1

Given C = `"x"^3/3 - 20"x"^2 + 70 "x"`

Marginal cost (C_m) = `"dC"/"dx"`

∴ (C_m) = `(3"x"^2)/3 - 20(2"x") + 70`

∴ (C_m) = x² - 40x + 70 

Differentiating w .r. t.x 

`("d"("C"_"m"))/"dx" = 2"x" - 40`

Differentiating w .r. t.x 

`("d"^2("C"_"m"))/"dx" = 2 > 0`

∴ C_m is minimum if 

`("d"("C"_"m"))/"dx"^2 = 0` and `("d"^2("C"_"m"))/"dx"^2 > 0`

∴ 2x - 40 = 0

`=> "x" = 40/2 = 20`

∴ x = 20

When x = 20 , `("d"^2("C"_"m"))/"dx"^2 = 2 > 0`

∴ C_m is minimum for x = 20