Answer to Question #315446 in Calculus for Jack Paul

Question

A firm's average revenue function 

𝐴𝑅=−18−7,5𝑄+𝑄

2

.

AR=−18−7,5Q+Q2.

Determine the number of units to be produced and sold to maximise revenue.

a.−1

b.6

c.3,75

d.0

Answer 1

$AR=\frac{R}{Q}\Rightarrow R=AR\cdot Q=\left( -18-7.5Q+Q^2 \right) Q=\\=Q^3-7.5Q^2-18Q\\Q^3-7.5Q^2-18Q\rightarrow \max \\3Q^2-15Q-18=0\Rightarrow Q\in \left\{ -1,6 \right\}$

Since Q is non-negative, we have only Q=6.

But this value is minimum, not maximum:

$R''\left( 6 \right) =\left( 6Q-15 \right) |_{Q=6}=36-15=21>0$

Actually the value of R tends to infinity at infinite Q.

So the problem is incorrect