What is the first derivative test for critical points?

1 Answer
Apr 22, 2018

If the first derivative of the equation is positive at that point, then the function is increasing. If it is negative, the function is decreasing.

Explanation:

If the first derivative of the equation is positive at that point, then the function is increasing. If it is negative, the function is decreasing.
See Also:
http://mathworld.wolfram.com/FirstDerivativeTest.html

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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#.

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