While crossing the Atlantic, sailors spot two mermaids 120° apart on each end of an island that is 6 miles away. How far apart a
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The distance between an arbitrary point on the surface and the origin is
Recall that for differentiable functions
and 
, the composition 
attains extrema at the same points that does, so we can consider an augmented distance function
The Lagrangian would then be
We have partial derivatives
Set each partial derivative to 0 and solve the system to find the critical points.
From the second equation it follows that either
or 
. In the first case we arrive at a contradiction (I'll leave establishing that to you). If , then we have
This means
so that the points on the surface closest to the origin are
.
Recall that for differentiable functions
The Lagrangian would then be
We have partial derivatives
Set each partial derivative to 0 and solve the system to find the critical points.
From the second equation it follows that either
This means
so that the points on the surface closest to the origin are