(a) Prove that for all [tex]$y \ge 0,$[/tex] there exists a unique real number [tex]$x$[/tex] such that
[tex]\[xe^x = y.\][/tex]
(b) By part (a), for [tex]$y \ge 0,$[/tex] we can let [tex]$f(y)$[/tex] be the unique real number such that
[tex]\[f(y) e^{f(y)} = y.\][/tex] Find
[tex]\[\int_0^e f(x) \ dx.\][/tex]
