A manufacturing company produces x items at a total cost of ₹ 40 + 2x. Their price per item is given as p = 120 – x. Find the value of x for which profit is increasing Solution: Total cost C = 40 + 2 - Mathematics and Statistics

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A manufacturing company produces x items at a total cost of ₹ 40 + 2x. Their price per item is given as p = 120 – x. Find the value of x for which profit is increasing

Solution: Total cost C = 40 + 2x and Price p = 120 − x

Profit π = R – C

∴ π = `square`

Differentiating w.r.t. x,

`("d"pi)/("d"x)` = `square`

Since Profit is increasing,

`("d"pi)/("d"x)` > 0

∴ Profit is increasing for `square`

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Solution

Total cost C = 40 + 2x and Price p = 120 − x

Profit π = R – C

∴ π = 120x – x2 – (40 + 2x)

= 120x – x2 – 40 – 2x

= – x2 + 118x – 40  

Differentiating w.r.t. x,

`("d"pi)/("d"x)` = – 2x + 118 

Since Profit is increasing,

`("d"pi)/("d"x)` > 0

∴ – 2x + 118 > 0

∴ 2x < 118

∴x < 59

∴ Profit is increasing for x < 59.

Concept: Application of Derivatives to Economics
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Chapter 1.4: Applications of Derivatives - Q.6

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