If a*b*c=8 then what is minimum value of (2+a)(2+b)(2+c) [closed]
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If $abc=8$ and $a,b,c >0$, then what is minimum possible value of $(2+a)(2+b)(2+c)$?
Edit: I got the answer and have posted it below.
inequality optimization
edited Nov 16 at 12:32
user21820
38k541150
asked Nov 15 at 10:21
Harry Potter
497
closed as off-topic by amWhy, José Carlos Santos, Dietrich Burde, Davide Giraudo, Christopher Nov 15 at 14:20
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, José Carlos Santos, Dietrich Burde, Davide Giraudo, Christopher
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
up vote
1
down vote
favorite
If $abc=8$ and $a,b,c >0$, then what is minimum possible value of $(2+a)(2+b)(2+c)$?
Edit: I got the answer and have posted it below.
inequality optimization
edited Nov 16 at 12:32
user21820
38k541150
asked Nov 15 at 10:21
Harry Potter
497
closed as off-topic by amWhy, José Carlos Santos, Dietrich Burde, Davide Giraudo, Christopher Nov 15 at 14:20
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, José Carlos Santos, Dietrich Burde, Davide Giraudo, Christopher
If this question can be reworded to fit the rules in the help center, please edit the question.
What have you tried? AM-GM inequality?
– астон вілла олоф мэллбэрг
Nov 15 at 10:23
@астонвіллаолофмэллбэрг I didn't tried that actually it didn't click me.
– Harry Potter
Nov 15 at 10:25
I got answer,thanks
– Harry Potter
Nov 15 at 10:26
2
If you have the answer, please write it down below and accept it.
– астон вілла олоф мэллбэрг
Nov 15 at 10:26
1
You must have some bounds otherwise a=b=c=2 is good because the expression is always increasing positive.
– NoChance
Nov 15 at 10:26
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
If $abc=8$ and $a,b,c >0$, then what is minimum possible value of $(2+a)(2+b)(2+c)$?
Edit: I got the answer and have posted it below.
inequality optimization
edited Nov 16 at 12:32
user21820
38k541150
asked Nov 15 at 10:21
Harry Potter
497
If $abc=8$ and $a,b,c >0$, then what is minimum possible value of $(2+a)(2+b)(2+c)$?
Edit: I got the answer and have posted it below.
inequality optimization
inequality optimization
edited Nov 16 at 12:32
user21820
38k541150
asked Nov 15 at 10:21
Harry Potter
497
edited Nov 16 at 12:32
user21820
38k541150
asked Nov 15 at 10:21
Harry Potter
497
edited Nov 16 at 12:32
user21820
38k541150
edited Nov 16 at 12:32
user21820
38k541150
edited Nov 16 at 12:32
user21820
38k541150
38k541150
asked Nov 15 at 10:21
Harry Potter
497
asked Nov 15 at 10:21
Harry Potter
497
asked Nov 15 at 10:21
Harry Potter
497
497
closed as off-topic by amWhy, José Carlos Santos, Dietrich Burde, Davide Giraudo, Christopher Nov 15 at 14:20
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, José Carlos Santos, Dietrich Burde, Davide Giraudo, Christopher
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by amWhy, José Carlos Santos, Dietrich Burde, Davide Giraudo, Christopher Nov 15 at 14:20
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, José Carlos Santos, Dietrich Burde, Davide Giraudo, Christopher
If this question can be reworded to fit the rules in the help center, please edit the question.
What have you tried? AM-GM inequality?
– астон вілла олоф мэллбэрг
Nov 15 at 10:23
@астонвіллаолофмэллбэрг I didn't tried that actually it didn't click me.
– Harry Potter
Nov 15 at 10:25
I got answer,thanks
– Harry Potter
Nov 15 at 10:26
2
If you have the answer, please write it down below and accept it.
– астон вілла олоф мэллбэрг
Nov 15 at 10:26
1
You must have some bounds otherwise a=b=c=2 is good because the expression is always increasing positive.
– NoChance
Nov 15 at 10:26
add a comment |
What have you tried? AM-GM inequality?
– астон вілла олоф мэллбэрг
Nov 15 at 10:23
@астонвіллаолофмэллбэрг I didn't tried that actually it didn't click me.
– Harry Potter
Nov 15 at 10:25
I got answer,thanks
– Harry Potter
Nov 15 at 10:26
2
If you have the answer, please write it down below and accept it.
– астон вілла олоф мэллбэрг
Nov 15 at 10:26
1
You must have some bounds otherwise a=b=c=2 is good because the expression is always increasing positive.
– NoChance
Nov 15 at 10:26
What have you tried? AM-GM inequality?
– астон вілла олоф мэллбэрг
Nov 15 at 10:23
What have you tried? AM-GM inequality?
– астон вілла олоф мэллбэрг
Nov 15 at 10:23
@астонвіллаолофмэллбэрг I didn't tried that actually it didn't click me.
– Harry Potter
Nov 15 at 10:25
@астонвіллаолофмэллбэрг I didn't tried that actually it didn't click me.
– Harry Potter
Nov 15 at 10:25
I got answer,thanks
– Harry Potter
Nov 15 at 10:26
I got answer,thanks
– Harry Potter
Nov 15 at 10:26
2
2
If you have the answer, please write it down below and accept it.
– астон вілла олоф мэллбэрг
Nov 15 at 10:26
If you have the answer, please write it down below and accept it.
– астон вілла олоф мэллбэрг
Nov 15 at 10:26
1
1
You must have some bounds otherwise a=b=c=2 is good because the expression is always increasing positive.
– NoChance
Nov 15 at 10:26
You must have some bounds otherwise a=b=c=2 is good because the expression is always increasing positive.
– NoChance
Nov 15 at 10:26
add a comment |
2 Answers
2
active
oldest
votes
up vote
3
down vote
I saw comment of A.M., G.M. inequality and solved it on my own. just posting answer
answered Nov 15 at 10:46
Harry Potter
497
3
Can you render it into text? Thanks.
– Oscar Lanzi
Nov 15 at 10:50
How could this lead to values for a,b and c?
– NoChance
Nov 15 at 10:52
You still have to show in your thesis that number $64$ is the minimum, i.e there exist $a,b,c$ such that....
– dmtri
Nov 15 at 17:34
add a comment |
up vote
2
down vote
You can solve this directly using the method of Lagrange multipliers: the critical points are solutions to
begin{align*}
(2+a)(2+b) + lambda a b &=0\
(2+a)(2+c) + lambda a c &=0\
(2+b)(2+c) + lambda b c &=0\
abc &= 8,
end{align*}
and eliminating equations gives you $a=b=c=2$.
You don't need to consider the case where any of your inequality constraints are active, since your objective function diverges as, say, $ato0$.
answered Nov 15 at 10:42
user7530
34.3k759112
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
I saw comment of A.M., G.M. inequality and solved it on my own. just posting answer
answered Nov 15 at 10:46
Harry Potter
497
3
Can you render it into text? Thanks.
– Oscar Lanzi
Nov 15 at 10:50
How could this lead to values for a,b and c?
– NoChance
Nov 15 at 10:52
You still have to show in your thesis that number $64$ is the minimum, i.e there exist $a,b,c$ such that....
– dmtri
Nov 15 at 17:34
add a comment |
up vote
3
down vote
I saw comment of A.M., G.M. inequality and solved it on my own. just posting answer
answered Nov 15 at 10:46
Harry Potter
497
3
Can you render it into text? Thanks.
– Oscar Lanzi
Nov 15 at 10:50
How could this lead to values for a,b and c?
– NoChance
Nov 15 at 10:52
You still have to show in your thesis that number $64$ is the minimum, i.e there exist $a,b,c$ such that....
– dmtri
Nov 15 at 17:34
add a comment |
up vote
3
down vote
up vote
3
down vote
I saw comment of A.M., G.M. inequality and solved it on my own. just posting answer
answered Nov 15 at 10:46
Harry Potter
497
I saw comment of A.M., G.M. inequality and solved it on my own. just posting answer
answered Nov 15 at 10:46
Harry Potter
497
answered Nov 15 at 10:46
Harry Potter
497
answered Nov 15 at 10:46
Harry Potter
497
answered Nov 15 at 10:46
Harry Potter
497
497
3
Can you render it into text? Thanks.
– Oscar Lanzi
Nov 15 at 10:50
How could this lead to values for a,b and c?
– NoChance
Nov 15 at 10:52
You still have to show in your thesis that number $64$ is the minimum, i.e there exist $a,b,c$ such that....
– dmtri
Nov 15 at 17:34
add a comment |
3
Can you render it into text? Thanks.
– Oscar Lanzi
Nov 15 at 10:50
How could this lead to values for a,b and c?
– NoChance
Nov 15 at 10:52
You still have to show in your thesis that number $64$ is the minimum, i.e there exist $a,b,c$ such that....
– dmtri
Nov 15 at 17:34
3
3
Can you render it into text? Thanks.
– Oscar Lanzi
Nov 15 at 10:50
Can you render it into text? Thanks.
– Oscar Lanzi
Nov 15 at 10:50
How could this lead to values for a,b and c?
– NoChance
Nov 15 at 10:52
How could this lead to values for a,b and c?
– NoChance
Nov 15 at 10:52
You still have to show in your thesis that number $64$ is the minimum, i.e there exist $a,b,c$ such that....
– dmtri
Nov 15 at 17:34
You still have to show in your thesis that number $64$ is the minimum, i.e there exist $a,b,c$ such that....
– dmtri
Nov 15 at 17:34
add a comment |
up vote
2
down vote
You can solve this directly using the method of Lagrange multipliers: the critical points are solutions to
begin{align*}
(2+a)(2+b) + lambda a b &=0\
(2+a)(2+c) + lambda a c &=0\
(2+b)(2+c) + lambda b c &=0\
abc &= 8,
end{align*}
and eliminating equations gives you $a=b=c=2$.
You don't need to consider the case where any of your inequality constraints are active, since your objective function diverges as, say, $ato0$.
answered Nov 15 at 10:42
user7530
34.3k759112
add a comment |
up vote
2
down vote
You can solve this directly using the method of Lagrange multipliers: the critical points are solutions to
begin{align*}
(2+a)(2+b) + lambda a b &=0\
(2+a)(2+c) + lambda a c &=0\
(2+b)(2+c) + lambda b c &=0\
abc &= 8,
end{align*}
and eliminating equations gives you $a=b=c=2$.
You don't need to consider the case where any of your inequality constraints are active, since your objective function diverges as, say, $ato0$.
answered Nov 15 at 10:42
user7530
34.3k759112
add a comment |
up vote
2
down vote
up vote
2
down vote
You can solve this directly using the method of Lagrange multipliers: the critical points are solutions to
begin{align*}
(2+a)(2+b) + lambda a b &=0\
(2+a)(2+c) + lambda a c &=0\
(2+b)(2+c) + lambda b c &=0\
abc &= 8,
end{align*}
and eliminating equations gives you $a=b=c=2$.
You don't need to consider the case where any of your inequality constraints are active, since your objective function diverges as, say, $ato0$.
answered Nov 15 at 10:42
user7530
34.3k759112
You can solve this directly using the method of Lagrange multipliers: the critical points are solutions to
begin{align*}
(2+a)(2+b) + lambda a b &=0\
(2+a)(2+c) + lambda a c &=0\
(2+b)(2+c) + lambda b c &=0\
abc &= 8,
end{align*}
and eliminating equations gives you $a=b=c=2$.
You don't need to consider the case where any of your inequality constraints are active, since your objective function diverges as, say, $ato0$.
answered Nov 15 at 10:42
user7530
34.3k759112
answered Nov 15 at 10:42
user7530
34.3k759112
answered Nov 15 at 10:42
user7530
34.3k759112
answered Nov 15 at 10:42
user7530
34.3k759112
34.3k759112
add a comment |
add a comment |
What have you tried? AM-GM inequality?
– астон вілла олоф мэллбэрг
Nov 15 at 10:23
@астонвіллаолофмэллбэрг I didn't tried that actually it didn't click me.
– Harry Potter
Nov 15 at 10:25
I got answer,thanks
– Harry Potter
Nov 15 at 10:26
2
If you have the answer, please write it down below and accept it.
– астон вілла олоф мэллбэрг
Nov 15 at 10:26
1
You must have some bounds otherwise a=b=c=2 is good because the expression is always increasing positive.
– NoChance
Nov 15 at 10:26