Find all the critical points of the function

[Dustobusto]

Homework Statement

f(x) = x ln(x)

The attempt at a solution

f'(x) = product rule, resulting in 1 + ln (x)

So by way of solving the problem, set 1 + ln (x) = 0

Now idealistically, find something in x that, when added to 1, equals zero.

Now here's a problem I have. How do I know when to keep searching for something in ln (x) and when to subtract one on both sides to make ln (x) = 1 ? And even when I get to that point, how to I move forward?

 

[Dick]

Homework Statement

f(x) = x ln(x)

The attempt at a solution

f'(x) = product rule, resulting in 1 + ln (x)

So by way of solving the problem, set 1 + ln (x) = 0

Now idealistically, find something in x that, when added to 1, equals zero.

Now here's a problem I have. How do I know when to keep searching for something in ln (x) and when to subtract one on both sides to make ln (x) = 1 ? And even when I get to that point, how to I move forward?

I'm not sure why you are so confused. You want to solve log(x)=(-1). Exponentiate both sides.

 

[Mark44]

Homework Statement

f(x) = x ln(x)

The attempt at a solution

f'(x) = product rule, resulting in 1 + ln (x)

So by way of solving the problem, set 1 + ln (x) = 0

Now idealistically, find something in x that, when added to 1, equals zero.

Now here's a problem I have. How do I know when to keep searching for something in ln (x) and when to subtract one on both sides to make ln (x) = 1 ? And even when I get to that point, how to I move forward?

Add -1 to both sides to get ln(x) = -1
Now solve for x.