Minimum value of f ( x ) = x log 2 </msub> &#x2061;<!-- ⁡ -->

deceptie3j

deceptie3j

Answered question

2022-06-12

Minimum value of f ( x ) = x log 2 x + ( 1 x ) log 2 ( 1 x )
What is the minimum value of the following function for 0 < x < 1? ? Here the base of logarithm is 2 .
f ( x ) = x log 2 x + ( 1 x ) log 2 ( 1 x )

Answer & Explanation

alisonhleel3

alisonhleel3

Beginner2022-06-13Added 23 answers

Changing the base you have log 2 x = ln x ln 2 . So you have
f ( x ) = 1 ln 2 ( x ln x + ( 1 x ) ln ( 1 x ) )
and
f ( x ) = 1 ln 2 ( ln x ln ( 1 x ) ) = 1 ln 2 ln ( x 1 x )
So f = 0 for x 1 x = 1 , i.e. x = 1 / 2

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