91Ó°ÊÓ

In this exercise, we are dealing with demand and revenue functions. These functions help businesses decide the best price and quantity of goods or services to maximize their income.

To start, let's establish how demand impacts revenue. Suppose a swimming club offers memberships at $$\$200$$ each. However, if more people join, they reduce the membership price by \$1 for each extra person beyond 100. Here, 'demand' refers to the number of memberships sold, while 'revenue' is the total income from those memberships.

Considering this club, if we denote the number of memberships as \(x\), the price per membership changes when \(x > 100\). Specifically, the price per membership can be described as \(200 - (x - 100)\), simplifying to \(200 - x + 100\) or \(300 - x\).

The revenue function becomes:
\[ R(x) = x (300 - x) \
R(x) = 300x - x^2 \]

This equation tells us how total revenue changes based on the number of memberships sold.

To find the maximum revenue, we can use differentiation. Differentiation helps find the rate at which things change, which is perfect for revenue functions that change with sales.

First, we need to find the first derivative of the revenue function. The first derivative tells us the slope of the revenue function at any given point. For our revenue function \(R(x) = 300x - x^2\), the first derivative is:
\( \frac{dR}{dx} = 300 - 2x \)

Next, setting the first derivative to zero helps find where the maximum revenue occurs because at these points, the slope is zero:
\[ 300 - 2x = 0 \]

Solving this equation for \(x\), we get:
\[ x = 150 \]

This means 150 memberships might be the number we need to maximize the revenue. But before concluding, a further step is essential.

We need to ensure that \(x = 150\) gives us the maximum revenue. For this, we use the second derivative test. The second derivative provides information about the concavity of the function. In simpler terms, it tells us if the curve opens upwards or downwards at a particular point.

For our revenue function \(R(x) = 300x - x^2\), we calculate the second derivative as:
\[ \frac{d^2R}{dx^2} = -2 \]

Since the second derivative \( \frac{d^2R}{dx^2} = -2 \) is negative, it indicates the revenue function is concave down at \(x = 150\). A concave downward shape signifies a maximum point.

Thus, the club should aim to sell 150 memberships to achieve the highest revenue. The negative second derivative assures us that at this number of memberships, the revenue is maximized.