What are examples of $f''(operatorname{critical})=0$ with local extrema?
$begingroup$
The second derivative test of critical points shows the type of extreme at the critical point:
$f''(operatorname{critical})>0$, then it's local minimum.
$f''(operatorname{critical})<0$, then it's local maximum.
$f''(operatorname{critical})=0$, it may or may not be local extreme.
I searched the web for examples of functions that have $f''(text{critical})=0$ and it's local extreme, but didn't find any.
What are examples of $f''(operatorname{critical})=0$ with local extrema ?
derivatives
edited Dec 24 '18 at 2:44
Namaste
1
asked Sep 18 '15 at 16:45
Mohamed MostafaMohamed Mostafa
554213
$endgroup$
add a comment |
$begingroup$
The second derivative test of critical points shows the type of extreme at the critical point:
$f''(operatorname{critical})>0$, then it's local minimum.
$f''(operatorname{critical})<0$, then it's local maximum.
$f''(operatorname{critical})=0$, it may or may not be local extreme.
I searched the web for examples of functions that have $f''(text{critical})=0$ and it's local extreme, but didn't find any.
What are examples of $f''(operatorname{critical})=0$ with local extrema ?
derivatives
edited Dec 24 '18 at 2:44
Namaste
1
asked Sep 18 '15 at 16:45
Mohamed MostafaMohamed Mostafa
554213
$endgroup$
add a comment |
$begingroup$
The second derivative test of critical points shows the type of extreme at the critical point:
$f''(operatorname{critical})>0$, then it's local minimum.
$f''(operatorname{critical})<0$, then it's local maximum.
$f''(operatorname{critical})=0$, it may or may not be local extreme.
I searched the web for examples of functions that have $f''(text{critical})=0$ and it's local extreme, but didn't find any.
What are examples of $f''(operatorname{critical})=0$ with local extrema ?
derivatives
edited Dec 24 '18 at 2:44
Namaste
1
asked Sep 18 '15 at 16:45
Mohamed MostafaMohamed Mostafa
554213
$endgroup$
The second derivative test of critical points shows the type of extreme at the critical point:
$f''(operatorname{critical})>0$, then it's local minimum.
$f''(operatorname{critical})<0$, then it's local maximum.
$f''(operatorname{critical})=0$, it may or may not be local extreme.
I searched the web for examples of functions that have $f''(text{critical})=0$ and it's local extreme, but didn't find any.
What are examples of $f''(operatorname{critical})=0$ with local extrema ?
derivatives
derivatives
edited Dec 24 '18 at 2:44
Namaste
1
asked Sep 18 '15 at 16:45
Mohamed MostafaMohamed Mostafa
554213
edited Dec 24 '18 at 2:44
Namaste
1
asked Sep 18 '15 at 16:45
Mohamed MostafaMohamed Mostafa
554213
edited Dec 24 '18 at 2:44
Namaste
1
edited Dec 24 '18 at 2:44
Namaste
1
edited Dec 24 '18 at 2:44
Namaste
1
1
asked Sep 18 '15 at 16:45
Mohamed MostafaMohamed Mostafa
554213
asked Sep 18 '15 at 16:45
Mohamed MostafaMohamed Mostafa
554213
asked Sep 18 '15 at 16:45
Mohamed MostafaMohamed Mostafa
554213
554213
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
The usual simple answer is $f(x)=x^4$, which is obviously non-negative, so $x=0$ is a minimum. You can use similar ideas to cook up other more complicated examples for more specific situations.
answered Sep 18 '15 at 16:47
ChappersChappers
56k74295
$endgroup$
add a comment |
$begingroup$
Try $f(x)=x^4$ (which has an extremum) and $g(x)=x^5$ (which doesn't have extremum).
answered Sep 18 '15 at 16:49
ArthurArthur
122k7122210
$endgroup$
add a comment |
$begingroup$
Consider $f(x)=x^4$, $f''(x)=12x^2$, thus $f''(0)=0$ but clearly the function has a minimum at $x=0$.
answered Sep 18 '15 at 16:49
MASLMASL
713313
$endgroup$
$begingroup$
You are welcome.
$endgroup$
– MASL
Sep 18 '15 at 16:55
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The usual simple answer is $f(x)=x^4$, which is obviously non-negative, so $x=0$ is a minimum. You can use similar ideas to cook up other more complicated examples for more specific situations.
answered Sep 18 '15 at 16:47
ChappersChappers
56k74295
$endgroup$
add a comment |
$begingroup$
The usual simple answer is $f(x)=x^4$, which is obviously non-negative, so $x=0$ is a minimum. You can use similar ideas to cook up other more complicated examples for more specific situations.
answered Sep 18 '15 at 16:47
ChappersChappers
56k74295
$endgroup$
add a comment |
$begingroup$
The usual simple answer is $f(x)=x^4$, which is obviously non-negative, so $x=0$ is a minimum. You can use similar ideas to cook up other more complicated examples for more specific situations.
answered Sep 18 '15 at 16:47
ChappersChappers
56k74295
$endgroup$
The usual simple answer is $f(x)=x^4$, which is obviously non-negative, so $x=0$ is a minimum. You can use similar ideas to cook up other more complicated examples for more specific situations.
answered Sep 18 '15 at 16:47
ChappersChappers
56k74295
answered Sep 18 '15 at 16:47
ChappersChappers
56k74295
answered Sep 18 '15 at 16:47
ChappersChappers
56k74295
answered Sep 18 '15 at 16:47
ChappersChappers
56k74295
56k74295
add a comment |
add a comment |
$begingroup$
Try $f(x)=x^4$ (which has an extremum) and $g(x)=x^5$ (which doesn't have extremum).
answered Sep 18 '15 at 16:49
ArthurArthur
122k7122210
$endgroup$
add a comment |
$begingroup$
Try $f(x)=x^4$ (which has an extremum) and $g(x)=x^5$ (which doesn't have extremum).
answered Sep 18 '15 at 16:49
ArthurArthur
122k7122210
$endgroup$
add a comment |
$begingroup$
Try $f(x)=x^4$ (which has an extremum) and $g(x)=x^5$ (which doesn't have extremum).
answered Sep 18 '15 at 16:49
ArthurArthur
122k7122210
$endgroup$
Try $f(x)=x^4$ (which has an extremum) and $g(x)=x^5$ (which doesn't have extremum).
answered Sep 18 '15 at 16:49
ArthurArthur
122k7122210
answered Sep 18 '15 at 16:49
ArthurArthur
122k7122210
answered Sep 18 '15 at 16:49
ArthurArthur
122k7122210
answered Sep 18 '15 at 16:49
ArthurArthur
122k7122210
122k7122210
add a comment |
add a comment |
$begingroup$
Consider $f(x)=x^4$, $f''(x)=12x^2$, thus $f''(0)=0$ but clearly the function has a minimum at $x=0$.
answered Sep 18 '15 at 16:49
MASLMASL
713313
$endgroup$
$begingroup$
You are welcome.
$endgroup$
– MASL
Sep 18 '15 at 16:55
add a comment |
$begingroup$
Consider $f(x)=x^4$, $f''(x)=12x^2$, thus $f''(0)=0$ but clearly the function has a minimum at $x=0$.
answered Sep 18 '15 at 16:49
MASLMASL
713313
$endgroup$
$begingroup$
You are welcome.
$endgroup$
– MASL
Sep 18 '15 at 16:55
add a comment |
$begingroup$
Consider $f(x)=x^4$, $f''(x)=12x^2$, thus $f''(0)=0$ but clearly the function has a minimum at $x=0$.
answered Sep 18 '15 at 16:49
MASLMASL
713313
$endgroup$
Consider $f(x)=x^4$, $f''(x)=12x^2$, thus $f''(0)=0$ but clearly the function has a minimum at $x=0$.
answered Sep 18 '15 at 16:49
MASLMASL
713313
answered Sep 18 '15 at 16:49
MASLMASL
713313
answered Sep 18 '15 at 16:49
MASLMASL
713313
answered Sep 18 '15 at 16:49
MASLMASL
713313
713313
$begingroup$
You are welcome.
$endgroup$
– MASL
Sep 18 '15 at 16:55
add a comment |
$begingroup$
You are welcome.
$endgroup$
– MASL
Sep 18 '15 at 16:55
$begingroup$
You are welcome.
$endgroup$
– MASL
Sep 18 '15 at 16:55
$begingroup$
You are welcome.
$endgroup$
– MASL
Sep 18 '15 at 16:55
add a comment |
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