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 Find the extremal curves
of the following constrained optimization problems, using the method of Lagrange multipliers:

, such that
.

, such that
, and
.

, such that
, and
.
 Suppose
and
are two points lying in the

plane, which is orientated vertically such that
is above
. Imagine there is a thin, flexible wire connecting the two points and lying entirely in the

plane. A frictionless bead travels down the wire, impelled by gravity alone. Show that the shape of the wire that results in the bead reaching the point
in the least amount of time is a cycloid, which takes the parametric form
where
is a constant.
 Find the curve
, in the interval
, which is of length
, and maximizes
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Previous: MultiFunction Variation
Richard Fitzpatrick
20160331