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