C(x)=2300+50x r(x)=830x-x^2 the maximum profit of the company is

maximum profit = $1,170,400

Explanation:

let

p(x) be the profit as a function of x (x - the number of items produced and sold)

the profit of a company, in dollars, is the difference between the company's revenue and cost

p(x) = r(x) - c(x)

p(x) = 830x - x² - 2300 - 50x

p(x) = - x²+ 780X - 2300

to determine the maximum profit of the company we need to find the maximum of the function p(x)

p(x) = - x²+ 780X - 2300 is quadratic function

since the quotient in front of x² is -1 < 0 the function has maximum in the vertex

for the function

f(x) = ax² + bx + c

the maximum value = c² - b² ÷ 4 a

in our case

a = -1

b = 780

c = - 2300

max p(x) = (-2300)² - (780)² ÷(4× (-1))

= (5290000 - 608400 ) ÷ (-4) =$ 1,170,400

The maximum profit is $1,170,400

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