Did 2 examples of constrained extremization of calculus problems: the first, understanding what dg = 0 means geometrically, and the second, an explicit example we could solve completely.
First, for geometric understanding, we considered the constrained
calculus problem of minimizing some function f(x,y,z) over 3-space,
subject to the constraint that 
. That is, we
only want to find those minima for f that lay in the plane obeying
y = x + c, that is the vertical plane (since z is arbitrary) that
contains the line y = x +c. We saw how taking dg=0 was equivalent
to restricting ourselves only to variations staying within the plane,
excluding those variations (dx, dy, dz) with any component
perpendicular to the plane. We then had to impose df = 0 (for an
extremum of f). Because dg=0 determines the relation -dx + dy =
0, df has two ways to be zero. The coefficients of dx and dy
can both be zero, independently; or dx and dy can have nonzero
coefficients, as long as they occur in the combination 
, which vanishes on the plane, where we are restricting our
attention. This possibility introduces new possible extrema, which
aren't extrema over the full 3-dimensional space, but are among the
extremal values that the function assumes on the plane. This is
because, with respect to all directions within the constrained surface
(the plane), the new extremal values are stationary; their only
non-stationary direction is normal to the plane.
We then did a more concrete example, maximizing f(x,y) = x+y, with
(x,y) constrained to lie on the unit circle. Again setting 
gave the points where f was stationary with respect to a
path along the circle; at these points f was nonstationary only in the
normal direction to the circle of allowed points. We had to solve for
obeying the constraint to find our two extremal points, the
intersection of the unit circle and the line 
.
We then generalized to calculus extremization problems with r
constraints 
. We noted that all 
vanish on the constrained
surface, so df can vanish by being any linear combination of the .
We introduced a sign by convention, setting 
. This means (reading off coefficients of the 
) that
.
Note that we could have gotten the same answer by minimizing
a different function, 
,
without any constraints. This is called the
Lagrange method of undetermined multipliers. Minimizing with respect
to 
reproduces the condition that 
be linearly
dependent on the 
. Minimizing with respect to
enforces the constraint that 
.
--
KB