Consider a consumer with a utility function(,) =6x²+defined over two goods, X and Y. Let the price of good X and the prince of good Y be 1 and let the consumer's income be 100.

a. Using differentials, find the marginal rate of substitution (which is the slope of an indifference curve) as a function ofand. Provide an economic interpretation for the marginal rate of substitution.

b. Set up the Lagrange problem and solve for the first order conditions.

c. Find the optimal consumption bundle.

d. Let the price of good X still be 1 but now consider a more general price for good Y given by. Also consider a general income level of M. How does the consumer's optimal choice ofandvary withand M?

e. Derive the indirect utility function for part (d) above.