Suppose ##f^{\prime\prime}## is continuous on an open interval that contains x = c(adsbygoogle = window.adsbygoogle || []).push({});

1. If ##f^{\prime}(c)=0## and ##f^{\prime\prime}(c)<0##, then ##f## has local maximum at x = c.

2. If ##f^{\prime}(c)=0## and ##f^{\prime\prime}(c)>0##, then ##f## has local minimum at x = c.

3. If ##f^{\prime}(c)=0## and ##f^{\prime\prime}(c)=0##, then the test fails. The function ##f## may have a local maximum, a local minimum, or neither.

I've a little doubt in point 3. I've come up with only 1 example for the possibility when

##f^{\prime}(c)=0##, ##f^{\prime\prime}(c)=0## and ##f## has neither local maximum or local minimum.

Ex: ##f(x)=x^3##

Please give examples for other two possibilities when

##f^{\prime}(c)=0##, ##f^{\prime\prime}(c)=0## and f has local maximum

##f^{\prime}(c)=0## , ##f^{\prime\prime}(c)=0## and f has local minimum

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# Second Derivative Test for Local Extrema

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