Critical Points of Log Function

In summary, the conversation discusses critical points in relation to a given function, with the key point being that the function is only defined for values of x greater than 0. The attempted solution suggests that x=0 and x=e are critical points, but the answer key only provides x=e as a critical point. The disagreement is resolved by noting that the function is not defined for x=0, making x=e the only critical point.
  • #1
Qube
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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.
 
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  • #2
the key to this problem is in the first line: "for all x > 0". Log(x) is not defined for any x ≤ 0
 
  • #3
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.
 

FAQ: Critical Points of Log Function

What is a critical point of a log function?

A critical point of a log function is any point where the derivative of the function is equal to zero or does not exist. This means that the slope of the function at that point is either a horizontal line or undefined.

How do you find critical points of a log function?

To find the critical points of a log function, you first need to take the derivative of the function. Then, set the derivative equal to zero and solve for x. The resulting values of x are the critical points of the function.

What is the significance of critical points in a log function?

Critical points are important in log functions because they can indicate where the function changes direction or has a maximum or minimum value. They can also be used to find the intervals where the function is increasing or decreasing.

Can there be more than one critical point in a log function?

Yes, there can be multiple critical points in a log function. This can occur when the function has multiple inflection points, where the concavity changes from positive to negative or vice versa.

How do critical points affect the graph of a log function?

Critical points can affect the graph of a log function by creating points of inflection, where the concavity changes. They can also indicate where the graph has maximum or minimum values, as well as where the function is increasing or decreasing.

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