Questions:

1.

Suppose that the functionf is given by

f(x)=x66x5+6x4+8

Show that fhas three critical points. Determine whether each critical point is a local maximum, local minimum or inflexion point.

Question 2

The functionf is defined by

f(x,y)=x2+7x+8xy+xy2+x2y

Find the critical points off and determine, for each, whether it is a local maximum, a local minimum or a saddle point.

Question 3

Express the following system of equations in matrix form, and solve it using row operations.

4xy3z=7

2x3yz=7

3x5y+z=16

Question 4

Determine the following integrals:

2x2+5x36x+4dx,(x+1)[ln(x2+2x+8)2]dx

Question 5

An arithmetic progression has the following properties:

- the sum of the first twelve terms is -840;

- the twelfth term is five times the second term.

Determine the first term and the common difference.

Question 6

The demand equation for a good isq(3p+2)=34 and the supply equation isq2p+8=0 wherep is the price andq is the quantity. Determine the equilibrium price and quantity. Sketch the supply and demand functions forp0.