Question

Problem 7.3. Let f (x, y) = x6 + 3xy + y2 + y4.
(a) Show that f remains unchanged if you replace x by −x and y by −y. Hence,
if (x, y) is a critical point of f, so is (−x, −y). Thus, critical points other than
(0, 0) come in ± pairs.
140 7 Optimization in Several Variables
(b) Compute the partial derivatives of f . Show that solve applied directly to
the system fx = fy = 0 fails to locate any of the critical points except for (0, 0).
(c) Let’s compensate by eliminating one of the variables and then using solve
followed by double. First solve for y in terms of x in the equation fx = 0.
Substitute back into the formula for fy and then apply first solve and then
double. You should end up with three critical values of x, giving a total of
three critical points. Find the numerical values of their coordinates. (Be sure
you have set x and y to be real; otherwise you will also end up with many
irrelevant complex critical points.)
(d) Confirm the calculation of the critical points by graphing the equations fx =
0 and fy = 0 on the same set of axes (using fimplicit and hold on). You
should see exactly one additional pair of critical points (in the sense of (a)).
(e) Classify the three critical points using the second derivative test.
(f) Apply fminsearch to f with the starting values (1, 1) and (0, 0). Show
that in the first case you go to a minimum and that in the second case you stay
near the saddle point.

Answer 1

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