## Extremum Test

Consider a function in 1-D. If has a relative extremum at , then either or is not Differentiable at . Either the first or second Derivative tests may be used to locate relative extrema of the first kind.

A Necessary condition for to have a Minimum (Maximum) at is

and

A Sufficient condition is and (). Let , , ..., , but . Then has a Relative Maximum at if is Odd and , and has a Relative Minimum at if is Odd and . There is a Saddle Point at if is Even.

See also Extremum, First Derivative Test, Relative Maximum, Relative Minimum, Saddle Point (Function), Second Derivative Test

© 1996-9 Eric W. Weisstein
1999-05-25