Question

a⁴+b⁴+c⁴>=abc(a+b+c)
Prove the above statement​

Answer 1

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Answer 2

Answ

From the  AM and GM inequality, we have

a  

4

+b  

4

≥2a  

2

b  

2

 

b  

4

+c  

4

≥2b  

2

c  

2

 

c  

4

+a  

4

≥2a  

2

c  

2

 

adding above inequalities and dividing by 2, we get

a  

4

+b  

4

+c  

4

≥a  

2

b  

2

+b  

2

c  

2

+c  

2

a  

2

......1

now we repeat the process of a  

2

b  

2

,b  

2

c  

2

 and c  

2

a  

2

 to get as below

a  

2

b  

2

+b  

2

c  

2

≥2b  

2

ac

b  

2

c  

2

+c  

2

a  

2

≥2c  

2

ab

c  

2

a  

2

+a  

2

b  

2

≥2a  

2

bc

adding the above and dividing by 2 we get

a  

2

b  

2

+b  

2

c  

2

+c  

2

a  

2

≥(b  

2

ac+c  

2

ab+a  

2

bc) or abc(b+c+a).....2

from (1) and (2) it follows

a  

4

+b  

4

+c  

4

≥abc(a+b+c)