I need a little help with finding absolute extrema bounded by a region.
Find the absolute extrema of over the region .
Therefore:
and
Where do I go from here and how?
By the Extreme Value Theorem, the extremes of given the constraint are either at points where or on the border of the region (i.e. where , since this is an ellipse). Let . Using the method of Lagrange Multipliers,
Substituting back into
we have
Hence the minimum value of is
and the maximum value of is .