Problem 1. Univariate unconstrained maximization. Consider the following maximization prob- lem: Let a > 1, and g(a) is a function of a max f(x; a) = a logr + (1 -a)x +g(a) IER 1. Write down the first order conditions for this problem with respect to r. 2. Solve explicitly for a* that satisfies the first order conditions. 3. Compute the second order conditions. Is the stationary point that you found in point 2 a maximum? Why (or why not)? 4. Is the function f concave in a? Problem 2. Multivariate unconstrained maximization. Consider the following maximization problem: max f (1, T2; a1, a2) = ,air] - all1 + ,a212 - 20212 + 2112 T1,12 1. Write down the first order conditions for this problem with respect to 21 and T2. 2. Solve explicitly for r; and x2 that satisfy the first order conditions. 3. Compute the second order conditions. Under what conditions for a, and a2 is the stationary point that you found in point 2 a maximum? 4. Under what conditions on a, and a2 is the function f concave in 1 and T2? When is it convex in 1 and x2?Problem 3. Multivariate differentiation. Consider a function: ay) = v32 392 where {3 E R is a known parameter. 1. Compute the rst derivative of f with respect to :1: and y. 2. Suppose now that y : e'z. What is the derivative of f (ageI) with respect to :12? Does it have any relationship with the derivatives in part (1) and the derivative of 6'\"? 3. Consider now two generic differentiable functions 9 : H.\"a } R and h : R > R. Consider g[:r, h(:r:)). What is the derivative of g