1.   For question II. 1 in the Module 1 problems, derive the second differential. Put the latter in matrix form.

2.   For question II.2 in the Module 1 problems, determine whether each stationary point is a local maximum, local minimum, or neither.

3.   Consider the function given in part (b.) of question II. 1 in the Module 1 problems.

a.) How many leading principal sub-matrices does the Hessian of the function have? Write

them out.

b.) What are the values of the leading principal minors?

c.) What shape is the function – positive definite, negative definite, or neither?

4.   For question II.3 in the Module 1 problems, prove that your recommended allocation of pages achieves a maximum of sales.

5.   A monopolist produces the same product in two factories, A and B. Total costs are as follows:

CA  = QA  + QA(2)   ,

CB  = 6QB  +1.5QB(2)   ,

where Q represents the quantity of the product. The total quantity is QT  = QA  + QB . The

inverse demand function for the product is

P = 56 − 2QT  .

What level of output should the monopolist produce in each factory in order to maximize profits? Prove that the solution you have found is actually a local maximum.

6.   Given the function f(x1 , x 2 ) = x1(2)  + x2(2) , find the stationary point(s) and determine whether it is a local maximum, local minimum, or neither under the following conditions.

a.)   0,   0

b.)   0,   0

a.)   and  have opposite signs