3. Implicit Functions and Implicit Differentiation. Consider a consumer with utility function u(X1,X2) = X1X2 3a. Let u" denote a particular utility level. Define x.(x. ) to be the level of x, such that bundle (x,,X, (x )) yields utility u". Explicitly derive x(x ) and find the derivative, dx,/dxj. How does this relate to the marginal rate of substitution? 3b. Write down an identity relating x, x, (x ) and u". Implicitly differentiate it and solve for dx,dx,. Compare your answer to the answer to part a. 3c. The Implicit Function Theorem (Simon and Blume, Theorem 15.1): Let G(x,y) be a continuously differentiable function such that G(x, y.) = c, and consider the expression G(x.y) = c. If G/by(X,.yo) = 0 then there exists a continuously differentiable function y = y(x) defined near (Xo yo) such that: (1) G(x,y(x)) = C (2) y(x ) = yo, and