ECON 11: Problem Set 1Due Thursday October 6 before 9:30am in class.1 Unconstrained optimization in one variableConsider the following function:f (x) = 3x3− 5x2+ xwherex ∈ [0,2]1. Find the x’s that are critical value points of f (x) (values of x that are a potential minimum or maximum).Are those critical values maxima or minima? (Hint: Check the second order derivative. Check also thevalue of the function at the endpoints 0 and 2 to make sure you have the correct answer.)2 Unconstrained optimization in two variablesConsider the function:f (x1,x2) = x31+ 3x32− 9x1x21. Find a minimum given that x1,x2≥ 1.3 Constrained optimizationJulia maximizes the following utility function:u(x1,x2) = x2/71x5/72subject to the budget constraintp1x1+ p2x2= I1wherep1, p2, x1, x2, I > 01. Find the (x1,x2) that maximizes u(x1,x2).2. Show that the maximizer x∗2(p1, p2,I) is decreasing in p2and increasing in I (Hint: Use the partial deriva-tives).3. Does x∗2(p1, p2,I) change if p1changes? (Hint: partial
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