Question

A company estimates that if x thousand cedis is spent on the marketing of a certain product, Q(x)
thousand units of the product will be sold, where Q(x) = 7x
27 + x
2
.
(a) Determine the domain of the function defined. [2 Marks]
(b) Determine how much of the product will be sold if the company does not spend any money on
marketing. [2 Marks]
(c) Determine the amount of money the company must spend on marketing to achieve the minimum
and maximum quantities. Leave your answers to 2 decimal places. [10 Marks]
(d) Determine the maximum and minimum quantities that can be sold. Leave your answer to the
nearest whole number. [5 Marks]
(e) Determine the marketing expendure ranges on which quantity sold is increasing and decreasing
and interpret your solution. [9 Marks]
(f) Determine what happens as the company invests an infintely large amount of money on market-
ing.

Answer 1

a) To find the domain we put denominator equal to 0

x = -3 underrot 3

So domain will be all values except -3 underrot 3

b) As x is showing market amount, so x = 0 . Substitute value of x as 0 in the given function

So it will be 1/27

c) x is maximised and minimised , we differentiate given function , we will get

f'(x) = 0

= 7(27 -x^2) / (27 +x^2)^2

x = 5.196

so at this x is maximised

for minimised we find f ''(x) we wil get x = 9

d) To find quantities

maximized substitute value of x as 5.196 we get Q = 674

minimised subsititute value of x = 9 we get Q = 63/108