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

Answer:

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

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

Step-by-step 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 $(8\,s, 10\,s)$.