# 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. - Mathematics and Statistics

Sum

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.

Concept: Maxima and Minima
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