Minimum of a logarithmic function

[brett812718]

Homework Statement

Find the maximum and minimum values of the function F(x)=ln(x/(x^2+1)) on the interval (0,10]

Homework Equations

d/dx[lnu]=u'/u

The Attempt at a Solution

f'(x)=(1-x^2)/[(1+x^2)x]
f'(x)=0 when 1-x^2=0
1=x^2
x=1 which is the maximum
f(1)=-.693
what do I use for the minimum since f(0)=-$$\infty$$ ? I can't write -$$\infty$$ because 0 is not included in the interval.

 

[Defennder]

Does the question say that both necessarily exist?

 

[brett812718]

no, it does not say

 

[Defennder]

If so, then you should be able to solve the question as you already have.

EDIT: Meant to say instead "as you already almost have". Just need to add a note.

 

[Nick89]

The function does not have a minimum value, since it is undefined at 0. You can go arbitrarily close to 0 and the function value will get arbitrarily small.

You cannot say that x=0.00001 is a minimum value with f(x) = -11.513... because for example x=0.0000001 has f(x) = -16.118... which is even smaller. You can keep doing this forever.
In other words, if you tell me a minimum value for the graph, I simply take an x that is a tiny bit closer to 0 than your x and my value will be even smaller, so your value was not a minimum.

EDIT
I guess you could say that the function (defined on (0,10]) has a local minimum value at x = 10 because it is an endpoint though. I don't know if you have to do this...

 

[Defennder]

Well I guess it's a matter of semantics as to whether endpoints count as local minima/maxima.

 

[Nick89]

Well I guess it's a matter of semantics as to whether endpoints count as local minima/maxima.

Not really. If the question explicitly states the interval as (0,10] then x=10 is an endpoint (x=0 is not) and this will always be a local extremum (unless the graph is y=constant), right?

For example, if you have a function (let's say f(x) = x^2) that defines the area of a piece of land, and the function is defined at [0,4], then you could ask the question 'what is the maximum area of the land'. If you do not use the endpoint as an extremum you would answer that there was no maximum area, which doesn't make sense...

 

[HallsofIvy]

To support Nick, suppose f(x)= x and the interval is (0, 1]. Then f(x) has a maximum, 1, on that interval but does NOT have a minimum. Conversely, if the interval is [0, 1), then f(x) has a minimum 0 on the interval but no maximum.

It is NOT a matter of "semantics".