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Difference between Relative and Absolute Extrema
The distinction between relative and absolute boils down to the distinction between an open region, disc, or ball, and a domain. Let’s say that at the point 
the value 
is greater than or equal to 
for all other values of (x, y) in an open region, disc, or ball containing . Then has a relative maximum at . Suppose instead that is greater than or equal to for all other values of (x, y) in the entire domain of f. Then has an absolute maximum at . Note that a point that satisfies both of these conditions is both a relative maximum and an absolute maximum at the same time. Sometimes a relative maximum is called a local maximum.
To find the extrema of a function of two variables we first find a critical point . The point is a critical point if 
and 
. Also, if 
or 
does ot exist, then is a critical point. Relative extrema occur only at critical points.
In the above examples we used algebraic means to determine whether critical points gave maximum or minimum values of the function. With more complicated functions we rely on the Second Partials test.
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