The gradient of a function is a tool for taking derivatives in any direction.
It records the maximum and minimum rates of increase of the function and the
directions in which they occur. It also records the directions of no increase.
Its unofficial nickname is "the Swiss Army Knife of Calculus"!
This week we will also take optimization to surfaces in space. We extend the
technique from Calculus I of critical points, boundary points, and the first and
second derivative tests.
The logic of optimization is the same as in Calculus I. Any continuous function
over a closed and bounded set must achieve both an absolute minimum and an
absolute maximum. The issue that we will face is that the examination of the
boundary will be more complicated than it was for a real-valued function of a
real variable.
Sometimes we search for the extreme values of a function restricted by one or
more conditions. In these cases the gradient once again comes to our rescue in
the method of Lagrange multipliers.
There is a cost for using the method of the Lagrange multipliers. We frequently
will need to solve systems of nonlinear equations, carefully tracking cases.