Find where increasing/decreasing, concavity, local extrema and inflection points for f(x)=ln/x

In summary, for the function f(x)=ln(x)/x, the derivative is 1-ln(x)/x^2 and the critical point is (e,1/e). There is no concavity, and the local maximum is at (e,1/e) with no local minimum. There are also no inflection points. The function increases on the interval (0, e) and decreases on the interval (e, positive infinity).
  • #1
Emma_011
4
0
Find where increasing/decreasing, concavity, local extrema and inflection points for f(x)=ln/x

So here is what I have so far:

The derivative is 1-ln(x)/x^2

Critical points are (e,1/e)

No concavity

Local max is also (e,1/e) (no local min)

no inflection points

Increase on (0, e) and decrease on (e, positive infinity)

Is this correct? I tried to do a graph to justify my work
 
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  • #2
Emma_011 said:
Find where increasing/decreasing, concavity, local extrema and inflection points for f(x)=ln x/x

The derivative is 1-ln(x)/x^2
Critical points are (e,1/e)
No concavity
Local max is also (e,1/e) (no local min)
Hi Emma, ☺

All correct so far.
Btw, how did you conclude that there is no concavity?

no inflection points
How did you reach that conclusion?
What is an inflection point exactly?

Increase on (0, e) and decrease on (e, positive infinity)

Correct.
 

1. What is the process for finding where a function is increasing or decreasing?

To find where a function is increasing or decreasing, you will need to take the derivative of the function and set it equal to zero. Then, solve for x to find the critical points. Plug these critical points into the original function to determine if the function is increasing or decreasing at those points.

2. How do you determine the concavity of a function?

To determine the concavity of a function, you will need to take the second derivative of the function and set it equal to zero. Then, solve for x to find the inflection points. Plug these inflection points into the second derivative to determine the concavity of the function at those points.

3. What are local extrema and how do you find them?

Local extrema are the highest or lowest points on a graph within a specific interval. To find them, you will need to take the derivative of the function and set it equal to zero. Then, solve for x to find the critical points. Plug these critical points into the second derivative to determine if they are local maxima or minima.

4. How do you find inflection points for a function?

Inflection points are points on a graph where the concavity changes. To find them, you will need to take the second derivative of the function and set it equal to zero. Then, solve for x to find the inflection points. Plug these inflection points into the second derivative to determine the concavity at those points.

5. Can a function have multiple inflection points?

Yes, a function can have multiple inflection points. This occurs when the concavity of the function changes multiple times within a given interval. To find these points, you will need to take the second derivative of the function and set it equal to zero. Then, solve for x to find all the inflection points within the interval.

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