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The demand function in economics is a mathematical expression that shows the relationship between the quantity demanded of a good and its price, alongside other factors like consumer income, price of related goods, and tastes or preferences. In the given exercise, the demand function is represented as \(p = 20 - 0.02x\), where \(p\) is the price per unit of the good, and \(x\) is the quantity demanded.

From this equation, we see that the price decreases as the quantity demanded increases, indicating an inverse relationship which is common in demand functions. Knowing the demand function is crucial for businesses as it helps to predict consumer behavior and pricing strategies. For students to understand this better, it may be helpful to visualize the equation graphically, with quantity on the x-axis and price on the y-axis, to see how demand changes with price variations.

Total Revenue (TR) is a financial metric that assesses the overall income generated from selling a certain quantity of goods or services. It is calculated by multiplying the price of the good by the quantity sold, expressed as \(R(x) = p(x) \times x\). In order to maximize total revenue, one needs to find the point at which an increase in the quantity sold does not lead to a proportionate increase in revenue.

Mathematically, this requires finding the maximum value of the TR function, often by taking its first derivative and setting it to zero, then solving for \(x\). In the context of the problem, after finding the derivative \(R'(x)\), and setting it to zero, the value of \(x\) that maximizes the TR is identified. Subsequently, this value is used to find the corresponding \(p\) that will yield maximum revenue. Understanding the process of total revenue maximization is essential for making strategic decisions about pricing and production levels.

Price elasticity of demand is a measure that illustrates the responsiveness of the quantity demanded of a good or service to a change in its price. Unit elasticity, or unitary elasticity, occurs when the percentage change in quantity demanded is equal to the percentage change in price, implying an elasticity of \(-1\). In economic terms, this indicates that the total revenue remains unchanged with variations in price.

In the exercise, after finding the value of \(x\) that maximizes total revenue, substituting this value back into the elasticity formula yields an elasticity of \(-1\), confirming that the demand is unit elastic at that point. For students, recognizing when a product has unit elasticity can be valuable as it implies that any change in price—up or down—will not affect the overall revenue. It is a critical concept for setting prices in a way that maintains revenue stability.