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