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Sum

A manufacturer can sell x items at a price of ₹ (280 - x) each .The cost of producing items is ₹ (x^{2} + 40x + 35) Find the number of items to be sold so that the manufacturer can make maximum profit.

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#### Solution

Selling price = x . (280 - x) = 280x - x^{2}

Cost price = x^{2} + 40x + 35

Let P be the profit.

∴ P = selling price - cost price

= 280 x -x^{2} - (x^{2} + 40x + 35)

= -2x^{2}+ 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.

Concept: Maxima and Minima

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