# Second Derivative Test for Local Extrema

1. Apr 5, 2014

### 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

Last edited by a moderator: Apr 18, 2014
2. Apr 5, 2014

3. Apr 5, 2014

### HallsofIvy

Staff Emeritus
For the first, $f(x)= 1- x^4$. For the second, $f(x)= x^4$.

4. Apr 5, 2014

### 22990atinesh

Thanx I get it