Consider the following utility maximization problem, in which the utility function of the consumer is given by:
 

The consumer income is , and  and . Find the utility-maximizing  and  when the budget constraint of the consumer may or may not bind. (Apply Kuhn-Tucker method).

In this problem, the consumer aims to maximize their utility function subject to a budget constraint. The utility function is given as \( U(x, y) = x^{0.5} y^{0.5} \), where \( x \) and \( y \) are the quantities of two goods consumed. The consumer's income is \( I = 100 \), and the prices of the goods are \( p_x = 2 \) for good \( x \) and \( p_y = 3 \) for good \( y \).

The budget constraint is given by the equation \( p_x x + p_y y = I \), or in this case, \( 2x + 3y = 100 \). The task is to determine the values of \( x \) and \( y \) that maximize the consumer's utility subject to this budget constraint, while considering the possibility that the budget constraint may or may not bind.

### Step 1: Setup the Lagrangian Function

To solve this problem using the Kuhn-Tucker method, we need to set up the Lagrangian function, which incorporates both the utility function and the budget constraint. The Lagrangian \( \mathcal{L}(x, y, \lambda) \) for this utility maximization problem is:

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