If profit P=.01(AF+20Ak)x - .01(Ak)x^2,
then the derivative P'=.01(AF+20Ak) - .02(Ak)x.
Set P'=0 and x=(AF+20Ak)/2(Ak) = F/2k + 10, which is the optimal markup.
If 20k > F, then F/2k < 10, so Xoptimal=F/2k + 10 < 20.
QED
The manager of a supermarket usually adds a markup of 20% to the wholesale prices of all the goods he sells. He reckons that he has a loyal core of F customers and that, if he lowers his markup to x% he will attract an extra k(20-x) customers from his rivals. Each week the average shopper buys goods whose wholesale value is $A.
Show that the manager can increase his profit by reducing his mark-up below 20% provided that 20k > F.
Now I am not sure of how to show for the bolded part in the question.
Workings:
.... So, as the mark up is lowered to x%:
Let total customers C = F + 20k - kx
Then total profit:
$ x/100 * A(F + 20k - kx)