Question

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 = ${\displaystyle \square }$

Differentiating w.r.t. x,

${\displaystyle \frac{\text{d}\pi }{\text{d}x}}$ = ${\displaystyle \square }$

Since Profit is increasing,

${\displaystyle \frac{\text{d}\pi }{\text{d}x}}$ > 0

∴ Profit is increasing for ${\displaystyle \square }$

Answer 1

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,

${\displaystyle \frac{\text{d}\pi }{\text{d}x}}$ = – 2x + 118 

Since Profit is increasing,

${\displaystyle \frac{\text{d}\pi }{\text{d}x}}$ > 0

∴ – 2x + 118 > 0

∴ 2x < 118

∴x < 59

∴ Profit is increasing for x < 59.