Question

For the cost function C = 2000 + 1800x - 75x² + x³ find when the total cost (C) is increasing and when it is decreasing.

Answer 1

Given = 2000 + 1800x - 75x² + x³ 

Differentiating with respect to 'x' we get,

`"dC"/"dx" = 1800 - 150x + 3x^2`

`"dC"/"dx"` = 0

⇒ 1800 - 150x + 3x² = 0

⇒ 3(x² - 50x + 600) = 0

⇒ x² - 50x + 600 = 0  ...(Divided by 3)

⇒ (x - 30)(x - 20) = 0 ....`{(600 = -30 xx -20),(- 50 = -30 -20):},`

⇒ x = 30, 20

The possible intervals are (0, 20) (20, 30) and (30, ∞)

Intervals	Sign of `"dC"/"dx"`	Nature of Function
(0, 20) say x = 10	1800 - 150(10) + 3(10)2 = 600 (Positive)	Increasing
(20, 30) say x = 25	1800 - 150(25) + 3(25)2 = - 75 (Negative)	Decreasing
(30, ∞) say x = 40	1800 - 150(40) + 3(40)2 = 600 (Positive)	Increasing

Hence, total cost is increasing in (0, 20) and (30, ∞) and decreasing in (20, 30).