




1. Suppose
you have a function of 4 variables, f (x, y, z, t) that you wish to find critical
points for, and you must obey two constraints g(x, y, z, t) = 0, and h (x, y,
z, t) = 0.
Deduce the conditions that such a point must obey.
2. Suppose you are constrained to move on a path in 3 dimensions and
the path is presented to you in parametric form.
What are the conditions for f to have a critical point?
3. Suppose you are constrained to move on a surface in 3 dimensions, which surface is presented to you in parametric form. What are the conditions for a critical point for f?