Second Derivative Test for Local Extrema

[22990atinesh]

Suppose ##f^{\prime\prime}## is continuous on an open interval that contains x = c

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

 

[UltrafastPED]

When condition (3) occurs you can just take another derivative ... see
http://mathworld.wolfram.com/SecondDerivativeTest.html

And see the references.

 

[HallsofIvy]

For the first, [itex]f(x)= 1- x^4[/itex]. For the second, [itex]f(x)= x^4[/itex].

 

[22990atinesh]

For the first, [itex]f(x)= 1- x^4[/itex]. For the second, [itex]f(x)= x^4[/itex].

Thanx I get it