Question

The Average Cost function associated with producing and marketing x units of an item is given by AC = `x + 5 + 36/x`.

1. Find the Total cost function.
2. Find the range of values of x for which Average Cost is increasing.

Answer 1

Give, AC(x) = `x + 5 + 36/x`

a. T.C(x) = x × AC

= `x(x + 5 + 36/x)`

= x² + 5x + 36

b. Given, AC = `x + 5 + 36/x`

`d/dx(AC) = 1 - 36/x^2`

AC increases, when `d/dx(AC) > 0`

`\implies 1 - 36/x^2 > 0`

`\implies 1 > 36/x^2`

`\implies` x² > 36

`\implies` x² – 36 > 0

`\implies` (x – 6)(x + 6) > 0

`\implies` x > 6 or x < – 6

As x is positive, AC increases when x > 6