a manufacturer charges $24 for headphones and has been selling 1000 of them per week. He estimated that for every $1 price reduction, 100 more headphones can be sold per week. What price should he charge for the headphones to maximize his total profit? Show your work.
Assume x is price reduction. Let construct function of profit: 24 - x -price per one headphone,
also 1000+100x -number of sold headphones
then profit function looks like:
f = (24 - x) (1000 + 100x )
f = -100 x^2 + 1400x + 24000
we should find the value of x when this function has a maximum. For that take the first derivative:
f'= -200 x + 1400 in the point f'=0 we have maximum, so