This project deals with the maxima of surfaces in 3-dimensional space. You will need to answer a series of questions and submit the results for assessment. For all figures you create using Matlab, you need to also submit the commands that you used to create the output. Working must be shown for all calculations done by hand.
 
 

PART A

1. Create and print a graph which clearly shows the intersection of the surfaces
 

                                                       (1)

                                                               (2)

over the domain .

2. By hand, substitute (2) into (1) and determine the maximum value of (1) subject to the constraint given by (2). (You should be able to check the accuracy of your calculation from the graph you produce above.)

3. In the  plane, plot on the same graph equation (2) and a contour diagram for (1) at the values  where  is the constrained maximum value found in part 2. Again, make this plot over the domain 

PART B

For functions of one variable it is impossible for a continuous function to have two local maxima and no local minimum. In this question we will see that this is not the case for functions of many variables.

1. Use the ezplot and hold on commands to make a contour diagram for the function
 

for the values z = -2, -0.5, -0.1, with domain -2 < x < 2, -1 < y < 3.

2. By hand, show that z has only two critical points (1, 2) and (-1, 0)
and they are both local maxima.

3. Plot the line y=x+1 on your contour diagram. (Note that the two critical points lie on this line.)

4. Now substitute y=x+1 into z(x, y) to obtain a new function of one variable w(x)=z(x, x+1). We already know that w has local maxima at
x= -1, 1. Determine the value of x between - 1 and 1 for which w has a local minimum. (HINT: you will not be able to find the solution
exactly. Use the xzero command in 10.2 of the Matlab notes.)

5. Determine the minimum value of w(x) and plot the contour for z(x, y) at this value. What do you notice about this curve and the line y=x+1? Explain why it is possible to travel along the line y=x+1 from the local maximum for z at (-1,0) to the other local maximum at (-1,2), without passing through a local minimum for z.