Answer:
a) [tex]C(x)[/tex] is increasing in two regions: (i) [tex](+\infty, 8\,s)[/tex] and (ii) [tex](10\,s,+\infty)[/tex].
b) [tex]C(x)[/tex] decreases in [tex](8\,s, 10\,s)[/tex].
Step-by-step explanation:
Let [tex]C(x) = x^{3}-27\cdot x^{2}+240\cdot x +850[/tex], where [tex]x[/tex] is the quantity of insecticide, measured in hundreds of liters, and [tex]C(x)[/tex] 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 [tex]C(x)[/tex] are decreasing and increasing by means of the first derivative and graphing tools. The first derivative of the function is:
[tex]C'(x) = 3\cdot x^{2}-54\cdot x+240[/tex] (1)
Please notice that regions where C(x) is increasing has [tex]C'(x) > 0[/tex], whereas [tex]C'(x) < 0[/tex] when [tex]C(x) < 0[/tex].
We notice that [tex]C(x)[/tex] is increasing in two regions: (i) [tex](+\infty, 8\,s)[/tex] and (ii) [tex](10\,s,+\infty)[/tex]. Besides, [tex]C(x)[/tex] decreases in [tex](8\,s, 10\,s)[/tex].
