If a , b , c and d are positive real numbers satisfying the expression: a

Sarai Davenport

Sarai Davenport

Answered question

2022-06-05

If a , b , c and d are positive real numbers satisfying the expression:
a 4 + b 4 + c 4 + d 4 = 4 a b c d
then, prove that a = b = c = d

Answer & Explanation

Leland Ochoa

Leland Ochoa

Beginner2022-06-06Added 25 answers

Using AM-GM with a 4 , b 4 , c 4 , d 4 , we get
a 4 + b 4 + c 4 + d 4 4 a 4 b 4 c 4 d 4 4
This means that
a 4 + b 4 + c 4 + d 4 4 a b c d
With equality if and only if a 4 = b 4 = c 4 = d 4 a = b = c = d
Alternatively, we can just transform your proof into an AM-GM one:
We use AM-GM on a 4 and b 4 , c 4 and d 4 , a 2 b 2 and c 2 d 2
a 4 + b 4 2 a 4 b 4 c 4 + d 4 2 c 4 d 4 a 2 b 2 + c 2 d 2 2 a 2 b 2 c 2 d 2
Summing up the first two and then using the third inequality we get,
a 4 + b 4 + c 4 + d 4 2 ( a 2 b 2 + c 2 d 2 ) 2 2 a b c d 4 a b c d
With equality if and only if a 4 = b 4 and a 2 b 2 = c 2 d 2 which means that a = b = c = d

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