Assume there is an internal solution to the utility maximization problem. Utility maximization occurs where the ratio of the marginal utility of $x_{1}$ to the marginal utility of $x_{2}$ equals the ratio of $p_{1}$ to $p_{2}$.

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In the context of consumer theory, an internal solution to the utility maximization problem implies that the consumer chooses a combination of goods $x_{1}$ and $x_{2}$ such that the marginal rate of substitution (MRS), which is the ratio of the marginal utility of $x_{1}$ to the marginal utility of $x_{2}$, equals the price ratio of $p_{1}$ to $p_{2}$. This condition ensures that the consumer is maximizing utility given their budget constraint.

Assume There Is an Internal Solution to the Utility Maximization x1x_{1}x1​

Assume There Is an Internal Solution to the Utility Maximization $x_{1}$