Hi everyone,

Stuck on the following proof, any help would be appreciated! (Calculus a Complete Course, Adams, 3rd Ed., Ch3 Ch Review, in case anyone has the text)

I *think* I have part a) ok

a) Show that the fuction f(x) = x^x is strictly increasing on [e^-1, if[

y = x^x

ln(y) = x*ln(x)

d/dx => 1/y * y' = x * 1/x + ln(x)

y' = y(1+ln(x)) = x^x *(1+ln(x))

if x = 1/e, ln(x) = -1,

therefore y' = y (1 -1)= 0

for n > e^-1

y' = n^n * (1+ln(n)) > 0

induction, k = n0, for k+1

y' = (k+1)^(k+1)*(1+ln(k+1)) > 0

Derivative is always greater than zero so strictly increasing

I am lost on b)

b) If g is the inverse function to f of part (a), show that

lim y-> inf g(y)ln(ln(y))/ln(y) = 1

hint: start with the eq. y = x^x and take the ln of both sides twice

y = x^x

ln(ln(y)) = ln(ln(x^x))

Now what? I spent ages trying to figure this out and got no where

Cheers?