This week we will investigate the concept of maxima of surfaces. Begin by plotting the surface

over the domain . From the graph make a guess for the maximum value of . Remember you can rotate the graph in order to get the best view. Suppose your guess is . You can check your estimate by plotting the plane . To do this, first type

>> hold on

and hit enter. Then type the command to plot the surface . You should now see two surfaces plotted together on the screen. From this graph you should be able to make an educated estimate for the values of , which correspond to the maximum value of . Before continuing you need to type

>> hold off

so the next graph is not superimposed on the previous.

Redo the above procedure over a smaller domain which includes your estimated values . Plotting over a smaller domain should allow you to make a more accurate estimate for the maximum value. Finally, calculate by hand the points where the gradient of the function is zero, and evaluate the function at these points. Compare this result with your estimate for the maximum.

Repeat the above problem for the surface

over the domain