When 'x' and 'y' are in direct proportion , then the relation between 'x' and 'y' is __________
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The distance between a point and the origin is
. But since
, both
and
have the same critical points, so we can consider instead the squared distance,
.
We're looking for the extrema of subject to
and
. The Lagrangian is
with critical points where the partial derivatives vanish:
From the first two equations, it follows that .
Then in the last two equations,
If , then
; if
, then
.
So there are two critical points, and
.
Let . We have a minimum distance of
and maximum distance of
.