
Optimization of Income
Im having a little trouble coming up with an equation to find maximum gross income.
A television company agrees to establish a community antenna in a new suburb at a charge of $25 each for 1000 subscribers or less. To encourage more subscribers, it is agreed to give a discount of 10 cents to each subscriber for every additional 10 subscribers in excess of 1000. For the company to make the most income what number of subscribers should it have?
Any help would be greatly appreciated!

Let s be the number of subscribers. Then the price is $25 minus 0.1 * (s1000)/10. So the revenue for the tv company is given by:
I assume you can find the maximum of this function by setting the derivative to zero. The answer is s=1750.
But that assumes the price is changing continuously with the number of subscribers, which is only an approximation. To get the exact answer, we should calculate some values around 1750. We can reduce the amount of work by recognizing that the answer must end in 9.
s=1739, R=30780.3
s=1749, R=30782.4
s=1759, R=30782.5
s=1769, R=30780.6
So the answer is s=1759.
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 Hollywood