Show combination of affine functions and logs has at most one zeroFor x > 0,...

Jamiya Weber

Jamiya Weber

Answered

2022-06-26

Show combination of affine functions and logs has at most one zero
For x > 0, let
f ( x ) = ( x + 2 ) l o g ( x ) ( x + 1 ) l o g ( x + 1 )
Can anybody show that the equation f ( x ) = 0 ( x > 0 ) has at most one solution.

Answer & Explanation

pheniankang

pheniankang

Expert

2022-06-27Added 22 answers

We compute
f ( x ) = 1 + 2 / x + log x 1 log ( x + 1 ) = 2 / x log ( 1 + 1 / x ) .
We want to show that this is positive. Putting y = 1 / x, we just need to show 2 y > log ( 1 + y ) for all positive y.
But these two functions are equal when y = 0 and the result is then clear by the concavity of the logarithm.

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