This function has at least six critical points in the indicated domain. Our goal in this activity is to locate precisely these critical points and to classify each as maximum, minimum, or saddle point. For this purpose, contour plots turn out to be more useful than graphs.

1. Make a contour plot of f over the domain [-3,3] x [-5,5], and identify a part of the domain that you think contains a local maximum or minimum. Explain what features of the contour plot indicate a local maximum or minimum.

2. Zoom in on your selected part of the contour plot until you can find a two-significant-digit (2SD) approximation to the coordinates of the critical point.

3. Return to the original domain, and identify another region that you think contains a saddle point. Explain what features of the contour plot indicate a saddle point.

4. Zoom in on this new region until you can find a 2SD approximation to the coordinates of this critical point.

It would be very time-consuming to zoom in on the contour plot to obtain, say, 5SD accuracy for all the critical points. Another approach is to calculate the partial derivatives for f and then solve the system of two equations in two unknowns obtained by setting both derivatives equal zero.

5. Calculate and display the partial derivatives for f_x and f_y. Explain why it is likely to be difficult to solve f_x = 0 and f_y = 0 for critical points.

6. Determine whether your helper application can solve these equations. If so, record at least six critical points. If not, don't despair -- just go on to the next Part.