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# Derivatives and Maximum Profit

2. DVD Player Company
A DVD player company's weekly production costs, denoted by C , is given by the
Expression

C=4x2+10x+30

where x is the selling price of DVD players produced, in pounds. The number of
DVD players sold each week is given by S= 400 -x and the weekly revenue is
given by R = Sx . The weekly profit, P , can be found by subtracting weekly costs
from the weekly revenue.

(a) Find an expression for R in terms of x only and hence show that
P= -5x2 + 390x- 30 .

(b) Find the value of x at which P takes its maximum value. What is the maximum
weekly profit?

(c) Copy and complete the following table.

X 0 10 25 35 45 55 80
P 6595

On graph paper plot the function P.
(d) By solving an equation, find the values of x for which P is at least £4500.

#### Solution Summary

Maximum profit is found using derivatives. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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