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Extremum Test

Consider a function f(x) in one dimension. If f(x) has a relative extremum at x_0, then either f^'(x_0)=0 or f is not differentiable at x_0. Either the first or second derivative tests may be used to locate relative extrema of the first kind.

A necessary condition for f(x) to have a minimum (maximum) at x_0 is

f^'(x_0)=0,

and

f^('')(x_0)>=0 (f^('')(x_0)<=0).

A sufficient condition is f^'(x_0)=0 and f^('')(x_0)>0 (f^('')(x_0)<0). Let f^'(x_0)=0, f^('')(x_0)=0, ..., f^((n))(x_0)=0, but f^((n+1))(x_0)!=0. Then f(x) has a local maximum at x_0 if n is odd and f^((n+1))(x_0)<0, and f(x) has a local minimum at x_0 if n is odd and f^((n+1))(x_0)>0. There is a saddle point at x_0 if n is even.

See also

Extremum, First Derivative Test, Local Maximum, Local Minimum, Saddle Point, Second Derivative Test

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Cite this as:

Weisstein, Eric W. "Extremum Test." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExtremumTest.html

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