"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)

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

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

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

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)

Thanks!

Finding the critical values is an important step in finding the absolute minimum or maximum of a function. In this case, you are correct that one critical value of the function f(x) = x ln(x) exists due to the domain restriction.

To find the critical values, we need to find where the derivative of the function is equal to zero or undefined. The derivative of f(x) is given by f'(x) = 1 + ln(x).

To find where the derivative is equal to zero, we set f'(x) = 0:

1 + ln(x) = 0

To solve this equation, we subtract 1 from both sides:

ln(x) = -1

Now, in order to isolate x, we can exponentiate both sides using e as the base:

e^(ln(x)) = e^(-1)

This simplifies to:

x = 1/e

Remember, since ln(x) is a logarithmic function, the natural logarithm, ln(x), is undefined for x ≤ 0. So, we need to check this critical value is within the domain of the function.

Since x = 1/e is positive (as e is a positive constant), it falls within the domain of the function. Therefore, x = 1/e is a critical value.

Now, to determine whether this critical value leads to an absolute minimum or maximum, we need to evaluate the function at the critical point and compare it to the function values at the endpoints of its domain.

The domain of the function f(x) = x ln(x) is (0, +∞), with x > 0. The critical point x = 1/e is within this interval.

To find the function value at the critical point, substitute x = 1/e into the function f(x):

f(1/e) = (1/e) ln(1/e)

Remember, ln(1/e) equals -1. Therefore, we have:

f(1/e) = (1/e) * (-1) = -1/e

The function value at the critical point x = 1/e is -1/e.

Next, we need to check the function values at the endpoints of the domain. Since x > 0, we can consider the limit as x approaches 0 and as x approaches positive infinity.

As x approaches 0 from the positive side, ln(x) approaches negative infinity. Multiplying by x, we get:

lim(x->0+) (x ln(x)) = 0

The limit as x approaches infinity can be evaluated by considering the behavior of ln(x). As x becomes larger and larger, ln(x) also increases without bound. Thus, multiplying by x, we have:

lim(x->∞) (x ln(x)) = +∞

Since the limit as x approaches infinity is unbounded, there is no maximum value for the function.

Comparing the values, we find that f(1/e) = -1/e is less than the function value at the endpoint x = 0. Therefore, the absolute minimum value of the function f(x) = x ln(x) is -1/e.

In conclusion, through finding the critical values and evaluating the function at those critical points as well as the endpoints of the domain, we determined that the absolute minimum value of the function f(x) = x ln(x) is -1/e.