For the demand function given in each,Find the following. a)...

For the demand function given in each,Find the following. a) The elasticity b) The elasticity at the given price, stating whether the demand is elastic or inelastic c) The value(s) of $x$ for which total revenue is $a$ maximum (assume that $x$ is in dollars) $$q=D(x)=\frac{3000}{x} ; \quad x=60$$

For the demand function given in each,Find the following.
a) The elasticity
b) The elasticity at the given price, stating whether the demand is elastic or inelastic
c) The value(s) of $x$ for which total revenue is $a$ maximum (assume that $x$ is in dollars)
$$q=D(x)=\frac{3000}{x} ; \quad x=60$$

Key Concepts

Demand Function

A demand function is a mathematical representation that shows the relationship between the price of a good and the quantity demanded by consumers. It encapsulates how changes in price affect consumer purchase behavior, forming the basis for analyses in microeconomics, market equilibrium, and consumer theory.

Price Elasticity of Demand

Price elasticity of demand is a measure used to show the responsiveness of the quantity demanded of a product to a change in its price. It is calculated as the percentage change in quantity demanded divided by the percentage change in price, and it helps in understanding whether changes in price will lead to proportionately larger or smaller changes in demand.

Elastic vs Inelastic Demand

This concept distinguishes goods based on their sensitivity to price changes. When demand is elastic, a small change in price leads to a large change in the quantity demanded (elasticity greater than one), whereas inelastic demand occurs when the quantity demanded changes very little even with a significant change in price (elasticity less than one). This classification is crucial for making pricing and revenue maximization decisions.

Total Revenue Maximization

Total revenue maximization involves determining the price level at which the product of price and quantity demanded is the highest. This concept uses calculus, specifically differentiation, to locate the critical points where revenue is optimized. Understanding this helps firms set prices that maximize their earnings.

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