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First Derivative Test

StationaryPoint

Suppose f(x) is continuous at a stationary point x_0.

1. If f^'(x)>0 on an open interval extending left from x_0 and f^'(x)<0 on an open interval extending right from x_0, then f(x) has a local maximum (possibly a global maximum) at x_0.

2. If f^'(x)<0 on an open interval extending left from x_0 and f^'(x)>0 on an open interval extending right from x_0, then f(x) has a local minimum (possibly a global minimum) at x_0.

3. If f^'(x) has the same sign on an open interval extending left from x_0 and on an open interval extending right from x_0, then f(x) has an inflection point at x_0.

See also

Extremum, Global Maximum, Global Minimum, Inflection Point, Local Extremum, Local Maximum, Local Minimum, Maximum, Minimum, Second Derivative Test, Stationary Point Explore this topic in the MathWorld classroom

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972.

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First Derivative Test

Cite this as:

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

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