A Very Fast-Growing Function

Let f(x) = xx

(A) To get a feel for how fast f(x) grows, compare f(x) with x2, x5, and ex when x=5,10,20.

x x2 x5 ex xx
5    
10    
20    

(B) You've seen how big f(x) can get. Now try exploring with your calculator -- plugging in various (positive) values for x, and try this time around to find an x that makes f(x) as small as possible. It is "obvious" what this x might be? ______.

(C) Use the identity A = eln(A) to re-write f(x):   f(x) = xx = ____________________.

(D) Use the identity ln(BC) = (C)(ln(B)) to re-write f(x) again:

  f(x) = ____________________.

(E) Find f'(x).





(F) For which value of x is f(x) the smallest? How do you know?










Copyleft notice: Copyright © 1997 (revised 2006) by Dr. Harel Barzilai. Non-profit educational use explicitly allowed and encouraged.
May not use for any other purpose without written permission