Question

The total cost function y for x units is given by y = 3x`((x+7)/(x+5)) + 5`. Show that the marginal cost decreases continuously as the output increases.

Answer 1

The total cost function, y = `3x((x+7)/(x+5)) + 5`

To prove the marginal cost decreases continuously as the output increase we should prove `"dy"/"dx"` is positive.

y = `3x((x+7)/(x+5)) + 5`

`= 3x (((x + 5) + 2)/(x + 5)) + 5`

`= 3x ((x + 5)/(x + 5) + 2/(x + 5)) + 5`

y = `3x(1 + 2/(x+ 5)) + 5`

y = `3 (x + (2x)/(x + 5)) + 5`

`"dy"/"dx" = 3 "d"/"dx" [x + (2x)/(x + 5)] + "d"/"dx" (5)`

`= 3 [1 + 2 "d"/"dx" (x/(x + 5))] + 0`

`= 3 [1 + 2(((x + 5)1 - x(1))/(x+5)^2)]`

`= 3 [1 + 2((x + 5 - x)/(x+5)^2)]`

`= 3 [1 + 2(5/(x + 5)^2)]`

`= 3 [1 + 10/(x+5)^2]`, which is positive.

∴ The marginal cost decreases continuously of the output increases.