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The point M0 with
coordinates (x0;y0) is called
the extremum point
(maximum or minimum) of
the function z = f (x, y), if the
value of the function at this
point is the largest or
smallest in comparison with
the nearest neighboring
values
...


The extremes of the function
are local!
In the domain of definition,
the function may have values
that are larger or smaller than
extremes
...


The points at which the
partial derivatives of the
function z = f (x, y) are zero or
do not exist are called critical
points
...


The equality to zero of the
partial derivatives of a
function of two variables or
their absence does not mean
that the function necessarily
has an extremum (the
condition is necessary but not
sufficient)
...

2

2

Find critical points, make an
assumption about the
existence of extremes of the
function
...


There is no extremum,
because at the nearest points,
the function may be greater or
less than its value at the
critical point
...
To clarify the
presence of extremes at these
points requires knowledge of
sufficient conditions!



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