Question

SUppose the total cost C(x) to manufacture a quantity x of insecticide (in hundreds of liters) is given by

C(x) = x^3-27x^2+240x+850.
a. Where is C(x) decreasing?
b. Where is C(x) increases?

Answer 1

a)
`C(x)` is increasing in two regions: (i)
`(+\infty, 8\,s)` and (ii)
`(10\,s,+\infty)`.

b)
`C(x)` decreases in
`(8\,s, 10\,s)`.

Explanation:

Let
`C(x) = x^(3)-27\cdot x^(2)+240\cdot x +850`, where
`x` is the quantity of insecticide, measured in hundreds of liters, and
`C(x)` is the total manufacturing cost as a function of the quantity of the insecticide, measured in US dollars. A possible approach to determine which regions of
`C(x)` are decreasing and increasing by means of the first derivative and graphing tools. The first derivative of the function is:

`C'(x) = 3\cdot x^(2)-54\cdot x+240` (1)

Please notice that regions where C(x) is increasing has
`C'(x) > 0`, whereas
`C'(x) < 0` when
`C(x) < 0`.

We notice that
`C(x)` is increasing in two regions: (i)
`(+\infty, 8\,s)` and (ii)
`(10\,s,+\infty)`. Besides,
`C(x)` decreases in
`(8\,s, 10\,s)`.