Let x 0 </msub> &gt; x 1 </msub> &gt; x 2 </msub> &gt;

realburitv4

realburitv4

Answered question

2022-05-20

Let x 0 > x 1 > x 2 > x 3 be any positive real numbers . What is the largest value of the real number k such that..
log x 0 x 1 1993 + log x 1 x 2 1993 + log x 2 x 3 1993 k log log x 0 x 3 1993
How to solve such problems, please suggest thanks....

Answer & Explanation

Dominique Holmes

Dominique Holmes

Beginner2022-05-21Added 10 answers

Let y i = l o g 1993 x i 1 x i
for i = 1 , 2 , 3. Then the given inequality may be written as
1 y 1 + 1 y 2 + 1 y 3 k y 1 + y 2 + y 3
By using arihtmetic - geometric mean inequality :
( 1 y 1 + 1 y 2 + 1 y 3 ) ( y 1 + y 2 + y 3 ) 3 ( 1 y 1 y 2 y 3 3 ) 3 ( y 1 y 2 y 3 3 ) = 9
Equality holds if and only if y 1 = y 2 = y 3 or x 0 , x 1 , x 2 , x 3 forms a geometric progression .
Hence maximum value of k is 9.

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