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

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.

#### 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
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