Question

"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)

Answer 1

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

Answer 2

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

Answer 3

Oh! So ln(x) can equal a negative number! I just didn't realize that. Thank you!

Answer 4

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)

Answer 5

Thanks!