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Question
A certain manufacturing concern has total cost function C = 15 + 9x - 6x2 + x3. Find x, when the total cost is minimum.
Solution
Given C = 15 + 9x - 6x2 + x3
Differentiating with respect to 'x' we get,
`"dC"/"dx" = 9 - 12x + 3x^2`
`=> "dC"/"dx"` = 0
⇒ 9 - 12x + 3x2 = 0
⇒ x2 - 4x + 3 = 0 (Divided by 3)
⇒ (x - 1)(x - 3) = 0
⇒ x = 1, 3
Now, `("d"^2"C")/("dx"^2)` = - 12 + 6x
When x = 3,
`("d"^2"C")/("dx"^2)` = -12 + 6(3) = - 12 + 18 = 6 > 0
∴ Total cost is minimum when x = 3
∴ x = 3
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