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In a Firm the Cost Function for Output X is Given as C = "X"^2/3 - 20"X"^2 + 70 "X". - Mathematics and Statistics

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Sum

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  (Cm) is minimum.

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Solution

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

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

∴ (Cm) = `(3"x"^2)/3 - 20(2"x") + 70`

∴ (Cm) = 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`

∴ Cm 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`

∴ Cm is minimum for x = 20

Concept: Maxima and Minima
  Is there an error in this question or solution?
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