Critical Points of Log Function

[Qube]

Homework Statement

http://i.minus.com/jCH20SF290QIb.png

Homework Equations

Critical point: when the derivative = 0 or the derivative fails to exist.

The Attempt at a Solution

I got x = 0 and x = e as critical points.

When x = e, the function becomes 0 / e, which = 0. Therefore, e is a critical point of f.

When x = 0, the function becomes 1/0, which = ∞. The derivative of ∞ does not exist, so wouldn't x = 0 be a critical point?

The answer key disagrees; the only critical point the key provides is x = e.

 

[blinktx411]

the key to this problem is in the first line: "for all x > 0". Log(x) is not defined for any x ≤ 0

 

[vela]

Actually, it's the fact that f is defined only for x>0 that matters. If the problem said f(x) = (ln x)/x for all x>10, f would have no critical points.