You find yourself in a strange undulating landscape given by the function z = f (x, y) = cos y − cos x, where z is the elevation.
1. Find all maxima, minima, and saddle points. What are the level curves for z = 0?
Graph this function.
You are now at the origin and wish to hike to the point (4π, 0, 0). You contemplate two rather different routes.
2. Your first route always keeps you at the same elevation. Determine such a route of minimal length. What is the length?
3. Your second route always moves along the gradient. Determine such a route of minimal length, assuming you start hiking in the positive x-direction. What is its length? If you cannot find an exact answer, determine an upper bound and a lower bound between which the actual length must lie.
4. Which route is the shorter—that of part (b), or (c)?
Appropriate pictures should be supplied throughout. Justify your answers.