Maximum - Can this be solved algebraically?

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SUMMARY

The discussion centers on finding the absolute extrema of the function f(x) = e^-x * ln(lnx). The user successfully derived the first derivative and identified a critical point at approximately x = 3.5, but struggled to isolate x algebraically from the equation 1/(xlnx) - ln(lnx) = 0. It was concluded that numerical methods, specifically Newton's Method, are necessary for solving this problem without graphing utilities.

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  • Experience with numerical methods, especially Newton's Method.
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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
 
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No, I don't think you can solve that without using numerical methods, such as a graphing calculator.
 
I found out today that we were supposed to use Newtons Method with a scientific calculator
 

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