Maximum - Can this be solved algebraically?

[turdferguson]

Homework Statement

Find the absolute extrema of f(x) = e^-x * ln(lnx)

The Attempt at a Solution

Ive successfully taken the first derivative and set it to zero. The problem is checking the sign of 1/(xlnx) - ln(lnx)

No matter how I try to manipulate this, I can't seem to isolate x. Its clear with a graphing calculator that f' changes from positive to negative at about 3.5, but this is supposed to be done without a graphing utility (if I could use one in my answer, I might as well have found the maximum from the start by graphing). Can this be set to zero and solved without a calculator?

Other forms include
1 - x*lnx*ln(lnx) = 0
lnx^(xlnx) - e = 0

 

[Dick]

No, I don't think you can solve that without using numerical methods, such as a graphing calculator.

 

[turdferguson]

I found out today that we were supposed to use Newtons Method with a scientific calculator