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How do I find range using AM-GM inequality
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I have been given a function, $$ f(x)= x^2+ dfrac9x^2 $$ and I been told to find the range of this function using AM-GM inequality only.
I was able to calculate minimum value of $f(x)$ which is $6$
$$ x^2 + dfrac9x^2 ≥ 6$$
and by setting $x^2 = dfrac9x^2$ I get min value as $6$
But how do I calculate maximum value for the range?
functions inequality
asked Jul 18 at 17:27
William
801214
add a comment |Â
up vote
0
down vote
favorite
I have been given a function, $$ f(x)= x^2+ dfrac9x^2 $$ and I been told to find the range of this function using AM-GM inequality only.
I was able to calculate minimum value of $f(x)$ which is $6$
$$ x^2 + dfrac9x^2 ≥ 6$$
and by setting $x^2 = dfrac9x^2$ I get min value as $6$
But how do I calculate maximum value for the range?
functions inequality
asked Jul 18 at 17:27
William
801214
Think about what happens for large $x$ values
– Isaac Browne
Jul 18 at 17:30
or when $x$ is close to $0$
– Henry
Jul 18 at 17:30
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have been given a function, $$ f(x)= x^2+ dfrac9x^2 $$ and I been told to find the range of this function using AM-GM inequality only.
I was able to calculate minimum value of $f(x)$ which is $6$
$$ x^2 + dfrac9x^2 ≥ 6$$
and by setting $x^2 = dfrac9x^2$ I get min value as $6$
But how do I calculate maximum value for the range?
functions inequality
asked Jul 18 at 17:27
William
801214
I have been given a function, $$ f(x)= x^2+ dfrac9x^2 $$ and I been told to find the range of this function using AM-GM inequality only.
I was able to calculate minimum value of $f(x)$ which is $6$
$$ x^2 + dfrac9x^2 ≥ 6$$
and by setting $x^2 = dfrac9x^2$ I get min value as $6$
But how do I calculate maximum value for the range?
functions inequality
asked Jul 18 at 17:27
William
801214
edited Jul 18 at 17:31
asked Jul 18 at 17:27
William
801214
asked Jul 18 at 17:27
William
801214
asked Jul 18 at 17:27
William
801214
801214
Think about what happens for large $x$ values
– Isaac Browne
Jul 18 at 17:30
or when $x$ is close to $0$
– Henry
Jul 18 at 17:30
add a comment |Â
Think about what happens for large $x$ values
– Isaac Browne
Jul 18 at 17:30
or when $x$ is close to $0$
– Henry
Jul 18 at 17:30
Think about what happens for large $x$ values
– Isaac Browne
Jul 18 at 17:30
Think about what happens for large $x$ values
– Isaac Browne
Jul 18 at 17:30
or when $x$ is close to $0$
– Henry
Jul 18 at 17:30
or when $x$ is close to $0$
– Henry
Jul 18 at 17:30
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
2
down vote
Because of $x^2$ and the positivity of $frac9x^2$, the function has no maximum value (it is unbounded above as $xtoinfty$ or $xto0$).
answered Jul 18 at 17:30
Parcly Taxel
33.6k136588
How do I know at what values of $x$ I need to check?
– William
Jul 18 at 17:38
@William Formally: show that $lim_xtoinfty(x^2+9/x^2)=+infty$. This should be easy to do by considering that $x^2$ also goes to $infty$ as $xtoinfty$.
– Parcly Taxel
Jul 18 at 17:40
what is the motivation for showing $f(x) to + infty$. I mean it could be anything, right?
– William
Jul 18 at 17:45
@William It can be as large as you like. That is the point. There is no finite maximum value.
– Parcly Taxel
Jul 18 at 17:46
So you mean I can directly include $infty$ in range in every AM-GM range problem?
– William
Jul 18 at 17:48
 |Â
show 1 more comment
up vote
1
down vote
No maximum exists there since $$f(x)>x^2$$which means that $f(x)$ can be arbitrarily large. Here is a sketch
answered Jul 18 at 17:39
Mostafa Ayaz
8,6023630
I don't think I'm gonna get a graph calculator in exam :/
– William
Jul 18 at 17:42
Then i also wrote the analytic reason
– Mostafa Ayaz
Jul 18 at 17:43
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
Because of $x^2$ and the positivity of $frac9x^2$, the function has no maximum value (it is unbounded above as $xtoinfty$ or $xto0$).
answered Jul 18 at 17:30
Parcly Taxel
33.6k136588
How do I know at what values of $x$ I need to check?
– William
Jul 18 at 17:38
@William Formally: show that $lim_xtoinfty(x^2+9/x^2)=+infty$. This should be easy to do by considering that $x^2$ also goes to $infty$ as $xtoinfty$.
– Parcly Taxel
Jul 18 at 17:40
what is the motivation for showing $f(x) to + infty$. I mean it could be anything, right?
– William
Jul 18 at 17:45
@William It can be as large as you like. That is the point. There is no finite maximum value.
– Parcly Taxel
Jul 18 at 17:46
So you mean I can directly include $infty$ in range in every AM-GM range problem?
– William
Jul 18 at 17:48
 |Â
show 1 more comment
up vote
2
down vote
Because of $x^2$ and the positivity of $frac9x^2$, the function has no maximum value (it is unbounded above as $xtoinfty$ or $xto0$).
answered Jul 18 at 17:30
Parcly Taxel
33.6k136588
How do I know at what values of $x$ I need to check?
– William
Jul 18 at 17:38
@William Formally: show that $lim_xtoinfty(x^2+9/x^2)=+infty$. This should be easy to do by considering that $x^2$ also goes to $infty$ as $xtoinfty$.
– Parcly Taxel
Jul 18 at 17:40
what is the motivation for showing $f(x) to + infty$. I mean it could be anything, right?
– William
Jul 18 at 17:45
@William It can be as large as you like. That is the point. There is no finite maximum value.
– Parcly Taxel
Jul 18 at 17:46
So you mean I can directly include $infty$ in range in every AM-GM range problem?
– William
Jul 18 at 17:48
 |Â
show 1 more comment
up vote
2
down vote
up vote
2
down vote
Because of $x^2$ and the positivity of $frac9x^2$, the function has no maximum value (it is unbounded above as $xtoinfty$ or $xto0$).
answered Jul 18 at 17:30
Parcly Taxel
33.6k136588
Because of $x^2$ and the positivity of $frac9x^2$, the function has no maximum value (it is unbounded above as $xtoinfty$ or $xto0$).
answered Jul 18 at 17:30
Parcly Taxel
33.6k136588
answered Jul 18 at 17:30
Parcly Taxel
33.6k136588
answered Jul 18 at 17:30
Parcly Taxel
33.6k136588
answered Jul 18 at 17:30
Parcly Taxel
33.6k136588
33.6k136588
How do I know at what values of $x$ I need to check?
– William
Jul 18 at 17:38
@William Formally: show that $lim_xtoinfty(x^2+9/x^2)=+infty$. This should be easy to do by considering that $x^2$ also goes to $infty$ as $xtoinfty$.
– Parcly Taxel
Jul 18 at 17:40
what is the motivation for showing $f(x) to + infty$. I mean it could be anything, right?
– William
Jul 18 at 17:45
@William It can be as large as you like. That is the point. There is no finite maximum value.
– Parcly Taxel
Jul 18 at 17:46
So you mean I can directly include $infty$ in range in every AM-GM range problem?
– William
Jul 18 at 17:48
 |Â
show 1 more comment
How do I know at what values of $x$ I need to check?
– William
Jul 18 at 17:38
@William Formally: show that $lim_xtoinfty(x^2+9/x^2)=+infty$. This should be easy to do by considering that $x^2$ also goes to $infty$ as $xtoinfty$.
– Parcly Taxel
Jul 18 at 17:40
what is the motivation for showing $f(x) to + infty$. I mean it could be anything, right?
– William
Jul 18 at 17:45
@William It can be as large as you like. That is the point. There is no finite maximum value.
– Parcly Taxel
Jul 18 at 17:46
So you mean I can directly include $infty$ in range in every AM-GM range problem?
– William
Jul 18 at 17:48
How do I know at what values of $x$ I need to check?
– William
Jul 18 at 17:38
How do I know at what values of $x$ I need to check?
– William
Jul 18 at 17:38
@William Formally: show that $lim_xtoinfty(x^2+9/x^2)=+infty$. This should be easy to do by considering that $x^2$ also goes to $infty$ as $xtoinfty$.
– Parcly Taxel
Jul 18 at 17:40
@William Formally: show that $lim_xtoinfty(x^2+9/x^2)=+infty$. This should be easy to do by considering that $x^2$ also goes to $infty$ as $xtoinfty$.
– Parcly Taxel
Jul 18 at 17:40
what is the motivation for showing $f(x) to + infty$. I mean it could be anything, right?
– William
Jul 18 at 17:45
what is the motivation for showing $f(x) to + infty$. I mean it could be anything, right?
– William
Jul 18 at 17:45
@William It can be as large as you like. That is the point. There is no finite maximum value.
– Parcly Taxel
Jul 18 at 17:46
@William It can be as large as you like. That is the point. There is no finite maximum value.
– Parcly Taxel
Jul 18 at 17:46
So you mean I can directly include $infty$ in range in every AM-GM range problem?
– William
Jul 18 at 17:48
So you mean I can directly include $infty$ in range in every AM-GM range problem?
– William
Jul 18 at 17:48
 |Â
show 1 more comment
up vote
1
down vote
No maximum exists there since $$f(x)>x^2$$which means that $f(x)$ can be arbitrarily large. Here is a sketch
answered Jul 18 at 17:39
Mostafa Ayaz
8,6023630
I don't think I'm gonna get a graph calculator in exam :/
– William
Jul 18 at 17:42
Then i also wrote the analytic reason
– Mostafa Ayaz
Jul 18 at 17:43
add a comment |Â
up vote
1
down vote
No maximum exists there since $$f(x)>x^2$$which means that $f(x)$ can be arbitrarily large. Here is a sketch
answered Jul 18 at 17:39
Mostafa Ayaz
8,6023630
I don't think I'm gonna get a graph calculator in exam :/
– William
Jul 18 at 17:42
Then i also wrote the analytic reason
– Mostafa Ayaz
Jul 18 at 17:43
add a comment |Â
up vote
1
down vote
up vote
1
down vote
No maximum exists there since $$f(x)>x^2$$which means that $f(x)$ can be arbitrarily large. Here is a sketch
answered Jul 18 at 17:39
Mostafa Ayaz
8,6023630
No maximum exists there since $$f(x)>x^2$$which means that $f(x)$ can be arbitrarily large. Here is a sketch
answered Jul 18 at 17:39
Mostafa Ayaz
8,6023630
answered Jul 18 at 17:39
Mostafa Ayaz
8,6023630
answered Jul 18 at 17:39
Mostafa Ayaz
8,6023630
answered Jul 18 at 17:39
Mostafa Ayaz
8,6023630
8,6023630
I don't think I'm gonna get a graph calculator in exam :/
– William
Jul 18 at 17:42
Then i also wrote the analytic reason
– Mostafa Ayaz
Jul 18 at 17:43
add a comment |Â
I don't think I'm gonna get a graph calculator in exam :/
– William
Jul 18 at 17:42
Then i also wrote the analytic reason
– Mostafa Ayaz
Jul 18 at 17:43
I don't think I'm gonna get a graph calculator in exam :/
– William
Jul 18 at 17:42
I don't think I'm gonna get a graph calculator in exam :/
– William
Jul 18 at 17:42
Then i also wrote the analytic reason
– Mostafa Ayaz
Jul 18 at 17:43
Then i also wrote the analytic reason
– Mostafa Ayaz
Jul 18 at 17:43
add a comment |Â
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Think about what happens for large $x$ values
– Isaac Browne
Jul 18 at 17:30
or when $x$ is close to $0$
– Henry
Jul 18 at 17:30