As the title suggests I am trying to prove that for all c ∈ [ 0 , ∞ ) there exists x ∈ R such that x e x = c.Now, the previous parts of this question imply that the way we are meant to do this is by using the intermediate value theorem. Then we can consider x e x = f ( x ), but this function is defined on R → R and the IVT requires that our domain be [ a , b ] with a , b ∈ R . So how exactly can we do this? Please note that this comes from an analysis course so calc I or II methods won't work here.

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As the title suggests I am trying to prove that for all   c  ∈  [  0  ,  ∞  ) there exists   x  ∈      R   such that   x      e    x    =  c.Now, the previous parts of this question imply that the way we are meant to do this is by using the intermediate value theorem. Then we can consider   x      e    x    =  f  (  x  ), but this function is defined on       R    →      R   and the IVT requires that our domain be   [  a  ,  b  ] with   a  ,  b  ∈      R  . So how exactly can we do this? Please note that this comes from an analysis course so calc I or II methods won&#039;t work here.

scapatofc · Accepted Answer

hintIf   c  =  0, we will take   x  =  0.So, let us assume that   c  &amp;gt;  0.the function   f  :  x  ↦  x      e    x   satisfies      lim          x      →              0        +              f  (  x  )  =  0    ;        lim          x      →      ∞        f  (  x  )  =  +  ∞So, there exist   δ  &amp;gt;  0 and   A  &amp;gt;  0 such that  a  ≤  δ    ⟹    f  (  a  )  &amp;lt;      c    2  and  b  ≥  A    ⟹    f  (  b  )  &amp;gt;  2  cNow, apply IVT to   g  :  x  ↦  f  (  x  )  −  cat the intervall   [  a  ,  b  ].

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