Consider the following utility function:

U = X₁X₂

Where X₁ and X₂ are quantities consumed of two goods. You are considering the actions of a consumer that maximizes utility. He or she has a fixed income m, and faces prices P₁ and P₂. The consumer spends all of its income on the two commodities.

A. Assuming that the consumer maximizes utility, what would be the exact condition showing the ratio of each good purchased by the consumer?

B. Please derive the demand functions for X₁ and X₂., providing an algebraic expression for each and a diagrammatic representation.

C. What is the price elasticity of demand for commodity 1? What is the price elasticity of demand for commodity 2?

For all of these questions, please provide an answer as well as a derivation or explanation for your result.