Inequality: $7a+5b+12able9$ [closed]

Clash Royale CLAN TAG#URR8PPP

up vote
7
down vote

favorite

If we assume that $a,b$ are real numbers such that $9a^2+8ab+7b^2le 6$, how to prove that :

$$7a+5b+12able9$$

inequality

share|cite|improve this question

edited Jul 23 '15 at 17:01

Paolo

5791320

asked Sep 15 '12 at 11:19

rab

371

closed as off-topic by Daniel W. Farlow, Did, Davide Giraudo, Daniel Robert-Nicoud, user223391 Jul 23 '15 at 22:09

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." – Daniel W. Farlow, Did, Davide Giraudo, Daniel Robert-Nicoud, Community

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Think geometry.
  – darko
  Sep 15 '12 at 11:53
  

  
  
  
  

add a comment | 

up vote
7
down vote

favorite

If we assume that $a,b$ are real numbers such that $9a^2+8ab+7b^2le 6$, how to prove that :

$$7a+5b+12able9$$

inequality

share|cite|improve this question

edited Jul 23 '15 at 17:01

Paolo

5791320

asked Sep 15 '12 at 11:19

rab

371

closed as off-topic by Daniel W. Farlow, Did, Davide Giraudo, Daniel Robert-Nicoud, user223391 Jul 23 '15 at 22:09

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." – Daniel W. Farlow, Did, Davide Giraudo, Daniel Robert-Nicoud, Community

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Think geometry.
  – darko
  Sep 15 '12 at 11:53
  

  
  
  
  

add a comment | 

up vote
7
down vote

favorite

up vote
7
down vote

favorite

If we assume that $a,b$ are real numbers such that $9a^2+8ab+7b^2le 6$, how to prove that :

$$7a+5b+12able9$$

inequality

share|cite|improve this question

edited Jul 23 '15 at 17:01

Paolo

5791320

asked Sep 15 '12 at 11:19

rab

371

If we assume that $a,b$ are real numbers such that $9a^2+8ab+7b^2le 6$, how to prove that :

$$7a+5b+12able9$$

inequality

share|cite|improve this question

edited Jul 23 '15 at 17:01

Paolo

5791320

asked Sep 15 '12 at 11:19

rab

371

share|cite|improve this question

share|cite|improve this question

edited Jul 23 '15 at 17:01

Paolo

5791320

edited Jul 23 '15 at 17:01

Paolo

5791320

edited Jul 23 '15 at 17:01

Paolo

5791320

5791320

asked Sep 15 '12 at 11:19

rab

371

asked Sep 15 '12 at 11:19

rab

371

asked Sep 15 '12 at 11:19

rab

371

371

closed as off-topic by Daniel W. Farlow, Did, Davide Giraudo, Daniel Robert-Nicoud, user223391 Jul 23 '15 at 22:09

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." – Daniel W. Farlow, Did, Davide Giraudo, Daniel Robert-Nicoud, Community

If this question can be reworded to fit the rules in the help center, please edit the question.

closed as off-topic by Daniel W. Farlow, Did, Davide Giraudo, Daniel Robert-Nicoud, user223391 Jul 23 '15 at 22:09

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." – Daniel W. Farlow, Did, Davide Giraudo, Daniel Robert-Nicoud, Community

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Think geometry.
  – darko
  Sep 15 '12 at 11:53
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Think geometry.
  – darko
  Sep 15 '12 at 11:53
  

  
  
  
  

Think geometry.
– darko
Sep 15 '12 at 11:53

Think geometry.
– darko
Sep 15 '12 at 11:53

add a comment | 

1 Answer
1

active

oldest

votes

up vote
10
down vote

We have

$$2(a-b)^2+7left(a-frac12right)^2 + 5left(b-frac12right)^2 geq 0$$

which is equivalent to

$$7a+5b+12ableq 9a^2+7b^2+8ab+3 leq 6+3=9$$

The motivation here is to search for equality case by solving the system of equation in real values $a,b$

beginequation*
begincases
7a+5b+12ab=9 \
9a^2+7b^2+8ab=6
endcases
endequation*

which yields $a=b=frac12$. Thus the factors $left(a-frac12right)^2$, $left(b-frac12right)^2$ and $(a-b)^2$ are in order.

share|cite|improve this answer

edited Jul 23 '15 at 17:24

Paolo

5791320

answered Sep 15 '12 at 13:10

Ajat Adriansyah

622316

• 
  
  
  

  
  

  
  
  
  
  
  

  
  How do you get $a = b = dfrac12$ from the system of equations above?
  – Paolo
  Jul 23 '15 at 17:38
  

  
  
  
  

add a comment | 

1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
10
down vote

We have

$$2(a-b)^2+7left(a-frac12right)^2 + 5left(b-frac12right)^2 geq 0$$

which is equivalent to

$$7a+5b+12ableq 9a^2+7b^2+8ab+3 leq 6+3=9$$

The motivation here is to search for equality case by solving the system of equation in real values $a,b$

beginequation*
begincases
7a+5b+12ab=9 \
9a^2+7b^2+8ab=6
endcases
endequation*

which yields $a=b=frac12$. Thus the factors $left(a-frac12right)^2$, $left(b-frac12right)^2$ and $(a-b)^2$ are in order.

share|cite|improve this answer

edited Jul 23 '15 at 17:24

Paolo

5791320

answered Sep 15 '12 at 13:10

Ajat Adriansyah

622316

• 
  
  
  

  
  

  
  
  
  
  
  

  
  How do you get $a = b = dfrac12$ from the system of equations above?
  – Paolo
  Jul 23 '15 at 17:38
  

  
  
  
  

add a comment | 

up vote
10
down vote

We have

$$2(a-b)^2+7left(a-frac12right)^2 + 5left(b-frac12right)^2 geq 0$$

which is equivalent to

$$7a+5b+12ableq 9a^2+7b^2+8ab+3 leq 6+3=9$$

The motivation here is to search for equality case by solving the system of equation in real values $a,b$

beginequation*
begincases
7a+5b+12ab=9 \
9a^2+7b^2+8ab=6
endcases
endequation*

which yields $a=b=frac12$. Thus the factors $left(a-frac12right)^2$, $left(b-frac12right)^2$ and $(a-b)^2$ are in order.

share|cite|improve this answer

edited Jul 23 '15 at 17:24

Paolo

5791320

answered Sep 15 '12 at 13:10

Ajat Adriansyah

622316

• 
  
  
  

  
  

  
  
  
  
  
  

  
  How do you get $a = b = dfrac12$ from the system of equations above?
  – Paolo
  Jul 23 '15 at 17:38
  

  
  
  
  

add a comment | 

up vote
10
down vote

up vote
10
down vote

We have

$$2(a-b)^2+7left(a-frac12right)^2 + 5left(b-frac12right)^2 geq 0$$

which is equivalent to

$$7a+5b+12ableq 9a^2+7b^2+8ab+3 leq 6+3=9$$

The motivation here is to search for equality case by solving the system of equation in real values $a,b$

beginequation*
begincases
7a+5b+12ab=9 \
9a^2+7b^2+8ab=6
endcases
endequation*

which yields $a=b=frac12$. Thus the factors $left(a-frac12right)^2$, $left(b-frac12right)^2$ and $(a-b)^2$ are in order.

share|cite|improve this answer

edited Jul 23 '15 at 17:24

Paolo

5791320

answered Sep 15 '12 at 13:10

Ajat Adriansyah

622316

We have

$$2(a-b)^2+7left(a-frac12right)^2 + 5left(b-frac12right)^2 geq 0$$

which is equivalent to

$$7a+5b+12ableq 9a^2+7b^2+8ab+3 leq 6+3=9$$

The motivation here is to search for equality case by solving the system of equation in real values $a,b$

beginequation*
begincases
7a+5b+12ab=9 \
9a^2+7b^2+8ab=6
endcases
endequation*

which yields $a=b=frac12$. Thus the factors $left(a-frac12right)^2$, $left(b-frac12right)^2$ and $(a-b)^2$ are in order.

share|cite|improve this answer

edited Jul 23 '15 at 17:24

Paolo

5791320

answered Sep 15 '12 at 13:10

Ajat Adriansyah

622316

share|cite|improve this answer

share|cite|improve this answer

edited Jul 23 '15 at 17:24

Paolo

5791320

edited Jul 23 '15 at 17:24

Paolo

5791320

edited Jul 23 '15 at 17:24

Paolo

5791320

5791320

answered Sep 15 '12 at 13:10

Ajat Adriansyah

622316

answered Sep 15 '12 at 13:10

Ajat Adriansyah

622316

answered Sep 15 '12 at 13:10

Ajat Adriansyah

622316

622316

• 
  
  
  

  
  

  
  
  
  
  
  

  
  How do you get $a = b = dfrac12$ from the system of equations above?
  – Paolo
  Jul 23 '15 at 17:38
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  How do you get $a = b = dfrac12$ from the system of equations above?
  – Paolo
  Jul 23 '15 at 17:38
  

  
  
  
  

How do you get $a = b = dfrac12$ from the system of equations above?
– Paolo
Jul 23 '15 at 17:38

How do you get $a = b = dfrac12$ from the system of equations above?
– Paolo
Jul 23 '15 at 17:38

add a comment |