Let a ϵ R + and b = exp ⁡ ( − a ) .What...

Patricia Bean

Patricia Bean

Answered

2022-07-20

Let a ϵ R + and b = exp ( a ) .
What is the range of { b 2 log ( b ) }? Does it range ( , + )?
How can I show b log ( b ) 1 b 1 ?

Answer & Explanation

Sandra Randall

Sandra Randall

Expert

2022-07-21Added 17 answers

Given that a>0
b 2 log ( b ) = ( e a ) 2 log ( e a ) = ( e a ) 2 ( a ) = a e 2 a
therefore you need to find the range of a e 2 a . Since a>0, you can think of this as a positive constant times e 2 a . This produces the range of ( 0 , )
Second question:
b log ( b ) 1 b 1 e a log ( e a ) 1 e a 1 a e a 1 e a 1 a e a 1 1
which simplifies to
a e a 1
or
e a a + 1
I would recommend applying Bernoulli's inequality. Alternatively, you could observe that
0 < e a = 1 + a + a 2 2 + a 3 3 ! + a 4 4 ! + a 5 5 ! +

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