Profit Maximization - Statement: > A manufacturer has been selling $1000$ television sets a week at $\$480 $ each. A market survey indicates that for each $\$11$ rebate offered to a buyer, the number of sets sold will increase by $110$ per week. Questions : > a) Find the function representing the revenue $R(x)$, where $x$ is the number of $\$11$ rebates offered. For this, I got $(110x+1000)(480-11x)$. Which is marked correct > b) How large rebate should the company offer to a buyer, in order to maximize its revenue? For this I got $17.27$. Which was incorrect. I then tried $840999$, the sum total revenue at optimized levels > c) If it costs the manufacturer $\$160$ for each television set sold and there is a fixed cost of $\$80000$, how should the manufacturer set the size of the rebate to maximize its profit? For this, I received an answer of $10$, which was incorrect.

For part b, the question is: for what $x$ is $R(x)=(110x+1000)(480-11x)$ maximized?

First, differentiate using the power rule and then set the derivative equal to zero:

$$R'(x)=110(480-11x)-11(110x+1000)=-2420x+41800$$

$$R'(x)=0 \Leftrightarrow -2420x+41800 \Leftrightarrow x\approx17.2$$

The rebate offered must therefore be $$\$17.2\cdot11=\$190\,\text{dollars}$$