1. Calculate the marginal rate of substitution for the following utility functions:
(a) u(x₁,x₂) = ax₁ + bx₂
(b) u(x1,x2) = 2√x₁ + x₂
(c) u(x₁,x₂) = ln(x₁) + x₂
(d) u(X₁, X₂) = x₁x₂
(e) u(x₁, x₂) = x₁^a x₂^b
(f) u(x₁,x₂) = x₁^a + x₂^a

2. Marie Curie has a utility function U(x, y)= max {x, 2y).
Note that the max{} function essentially says that the value of the function takes the highest of the values in the set (the set is denoted as the things inside the curly braces).
(a) Draw a graph, plotting the indifference curve along which U(x, y) = 10.
(b) Are Marie's preferences convex?

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