Question

A manufacturer knows that if x goods are demanded on a particular week, the total cost and revenue functions will be:𝐶(𝑥) = 14 + 3𝑥 𝑎𝑛𝑑 𝑅(𝑥) = 18𝑥 − 2𝑥 2 respectively.

i. Calculate the level of demand that will maximize profits (6 marks)

ii. Calculate the amount of profit that will be realized at this maximum point. (4 marks) 

Answer 1

i)

profit function:

$P(x)=𝑅(𝑥)-𝐶(𝑥)=18x-2x^2-14-3x=-2x^2+15x-14$

$P'(x)=-4x+15=0$

level of demand that will maximize profits:

$x=15/4=3.75$

ii)

amount of profit that will be realized at this maximum point:

$P_{max}(x)=-2\cdot3.75^2+15\cdot3.75-14=14.125$