Posted by Stuck on Monday, March 22, 2010 at 10:12pm.
"Find the absolute minimum value of the function f(x) = x ln(x)".
I know that one of the critical values is 1 (domain restriction). Am I right?
So I did the derivative, and got 1+ ln(x), but I cannot figure out the critical values, because ln(x) can't equal 1...
Am I doing something wrong??
(Answer is 1/e)

Calculus  another question  Reiny, Monday, March 22, 2010 at 10:20pm
Why can't lnx = 1 ??
You were correct to have
ln x = 1
by definition
e^1 = x
or
x = 1/e
now put that back ...
f(1/e) = (1/e)ln(1/e)
= (1/e)(1) = 1/e
so the minimum value of the function is 1/e and it occurs when x = 1/e 
Calculus  another question  drwls, Monday, March 22, 2010 at 10:25pm
The derivative of
The log of a quantity CAN be negative, and is if the number is between 0 1nd 1. You were thinking of the rule that you cannot take the log of a negative number.
f(x) = x lnx leads to
f'(x) = lnx + 1
That equals zero when
lnx = 1
Make both sides the same power of e and the equation will still be valid.
x = e^1
That is where the function is a minimum. The value of that minimum is
ln(e^1)*e^1 = e^1 
Calculus  another question  Stuck, Monday, March 22, 2010 at 10:28pm
Oh! So ln(x) can equal a negative number! I just didn't realize that. Thank you!

Calculus  another question  Reiny, Monday, March 22, 2010 at 10:35pm
look at the graph of y = lnx
the range is any real number, it is in the domain where x > 0
take ln (.5)
or
ln (.367879441) 
Calculus  another question  Stuck, Monday, March 22, 2010 at 10:43pm
Thanks!