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Question
A manufacturer can sell x items at a price of ₹ (280 - x) each .The cost of producing items is ₹ (x2 + 40x + 35) Find the number of items to be sold so that the manufacturer can make maximum profit.
Solution
Selling price = x . (280 - x) = 280x - x2
Cost price = x2 + 40x + 35
Let P be the profit.
∴ P = selling price - cost price
= 280 x -x2 - (x2 + 40x + 35)
= -2x2+ 240x - 35
`therefore "dP"/"dx" = -4"x" + 240` ...(I)
`"dP"/"dx" = 0 => -4"x" + 240 = 0`
`therefore "x" = 60`
Differentiating (I). w.r.t. x, again
`("d"^2"P")/("dx"^2) = -4`
`therefore ("d"^2"P")/("dx"^2)` at (x = 60) = -4< 0
`therefore` P is maximum at x = 60.
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