Word problem with differentiation..
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.
.... 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)