# The revenue function for a bicycle shop is given by r (x) = x - p(x) dollars, where x is the...

## Question:

The revenue function for a bicycle shop is given by {eq}r (x) = x - p(x) {/eq} dollars, where {eq}x {/eq} is the number of units sold and {eq}\displaystyle p(x) = 200 - 0.5 x {/eq} is the unit price. Find the maximum revenue.

## Maxima and Minima:

The Maxima and Minima of a function are found by taking the first derivative of the given equation and finding the value of variables for which the first derivative is zero. Then take the second derivative and find its sign for the value of variable identified previously. If the second derivative is negative, then the value will give a maxima, and if it is positive, then the value will give a maxima.

The revenue is given as the product of the number of units sold and unit price of each product. Thus, the revenue is:

\begin{align} r(x) = x \times p(x) = x(200 - 0.5 x) = 200x - 0.5x^2 \end{align}\\

The first derivative and the value of variable for which it is zero is:

$$r'(x) = 200 - x = 0 \\ x = 200 \\$$

The second derivative and its sign for {eq}x = 200 {/eq} is:

\begin{align} r''(x) = - 1 < 0 \end{align}\\

Thus, the revenue is maximized for {eq}x = 200 {/eq}. Thus, maximum revenue is;

\begin{align} r(200) = 200(200) - 0.5(200)^2 = 20000 \end{align}\\

Thus, the maximum revenue is \$20,000.