Question

The total cost of manufacturing x articles C = 47x + 300x² – x⁴ . Find x, for which average cost is decreasing

Answer 1

Given, total cost function is

C = 47x + 300x² – x⁴ 

Average cost C_A = `"C"/"A"`

∴ C_A = `(47 x + 300x^2 - x^4)/x`

= `(x(47 + 300x - x^3))/x`

∴ C_A = 47 + 300x – x³

∴ `"dC"_"A"/"dx"` = 0 + 300x – 3x³

= 300x – 3x³

= 3(100 – x²)

Since average cost C_A is a decreasing function `"dC"_"A"/"dx" < 0`

∴ 3(100 – x²) < 0

∴ 100 – x² < 0

∴ 100 < x² 

∴ x² > 100

∴ x > 10 or x < – 10

But x < – 10 is not possible    .....[∵ x > 0]

∴  x > 10   

∴ The average cost C_A is decreasing for x > 10.