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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 (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
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