Maxima, Minima and Saddle Points
then The second derivative test goes: if
and (1) f has a local maximum at (a, b) if
at (a, b).
and (2) f has a local minimum at (a, b) if
at (a, b) .
at (a, b) . (3) f has a saddle point at (a, b) if
at (a, b) . (4) the test is inconclusive at (a, b)if
Remark:
is a necessary condition The condition
for having an Extreme value at an interior point of the domain of a differentiable function.
Example:
Find the extreme values of
Solution:
And,
The only possibility is the origin
at (a, b) = (0, 0) can be denoted as :
(0, 0) is a local minimum .
Example:
Find the extreme values of
Solution:
And,
The only possibility is the origin
i.e., the origin is the only point where f might have an extreme value.
at (a, b) = (0, 0) can be denoted as :
.f has a saddle at (0, 0).
Example:
Find the extreme values of
Solution:
And,
i.e., the point (-2, -2) is the only point where f may taken on an extreme value.
at (a, b)=(-2, -2) can be denoted as:
.f has a local maximum at (-2, -2).
Exercise:
(1) Find the extreme values of
(2) Find the extreme values of
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