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 ?

Question

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 ?

Sandra Randall · Accepted Answer

Given that a&amp;gt;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&amp;gt;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        ≤  1which simplifies to  a  ≤      e    a    −  1or      e    a    ≥  a  +  1I would recommend applying Bernoulli&#039;s inequality. Alternatively, you could observe that  0  &amp;lt;      e    a    =  1  +  a  +            a      2        2    +            a      3              3      !        +            a      4              4      !        +            a      5              5      !        +  ⋯

Let a epsilon RR_+ and b=exp(−a). What is the range of {-b^2 log(b)}? Does it range (−oo,+oo)? How can I show (-b log(b))/(1−b) <= 1 ?

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Answer & Explanation

New Questions in Calculus 1