Question

A manufacturing company produces x items at the total cost of Rs (180 + 4x). The demand function of this product is P = (240 − x). Find x for which profit is increasing.

Answer 1

Total cost function (C) = 180 + 4x
Demand function (P) = 240 − x

Where x is the number of items produced.

Total revenue (R) = P × D
∴ R = x (240 − x) 
∴ R = 240x − x²

Profit function π = R − C 
∴ π = (240x − x²) − (180 + 4x)
∴ π = 240x − x² − 180 − 4x
∴ π = − x² + 236x − 180

Differentiating w.r.t.x,

∴ `"dπ"/"dx"` = − 2x + 236

Profit  π is increasing if `"dπ"/"dx"` > 0

i.e. if − 2x + 236 > 0

i.e. if 236 > 2x

i.e. if x < `236/2`

i.e. if x < 118

∴ The profit is increasing for x < 118.