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Derivatives and Maximum Profit

Author:
September 23rd, 2022

DVD Player Company
A DVD player company’s weekly production costs, denoted by C, is given by the expression:

C=4×2+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= -5×2 + 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.

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