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In page Microeconomics:
"The utility maximization problem is a constrained optimization problem in which an individual seeks to maximize utility subject to a budget constraint. Economists use the extreme value theorem to guarantee that a solution to the utility maximization problem exists. That is, since the budget constraint is both bounded and closed, a solution to the utility maximization problem exists. Economists call the solution to the utility maximization problem a Walrasian demand function or correspondence.[citation needed]
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