Is f(x) = (x+1)/x^2 increasing or decreasing for x>0

[MarkSA]

My question is:

If f(x)=(x+1)/x^2 and x>0, Is f(x) increasing or decreasing on (0,infinity)?
Justify your answer.

It's clearly increasing, but how can I prove that? If I differentiate it, I get (-x-2)/x^3. The critical numbers are 0 and -2, but since it was stated that x>0, i'm guessing I can't use the increasing/decreasing test to justify it. Or can I?

If not, what other methods might I use? Thanks.

 

[daon]

Re: Is f(x) increasing or decreasing

Hello,

My question is:
If f(x)=(x+1)/x^2 and x>0, Is f(x) increasing or decreasing on (0,infinity)?
Justify your answer.

It's clearly increasing, but how can I prove that? If I differentiate it, I get (-x-2)/x^3. The critical numbers are 0 and -2, but since it was stated that x>0, i'm guessing I can't use the increasing/decreasing test to justify it. Or can I?

If not, what other methods might I use? Thanks.

Why is it clearly increasing, did you graph it for yourself? If x>0 then what is the derivative, (-x-2)/x^3, going to be... positive, 0, or negative... and what does that tell you about it increasing/decreasing/etc.?

 

[MarkSA]

Oops, I see now. Sorry about that, I think i've been at this too long today. Thanks for your patience

 

[MarkSA]

However,

is it legitimate in the particular above case to use critical numbers in a function to find intervals of increase/decrease by testing a number on each side of the critical numbers to see if it is positive/negative? (Positive meaning increasing, negative decreasing on that interval by the I/D test)

It makes sense to me that it should be, since the increasing/decreasing of the function should be unchanged at the interval in question even if you shrink the domain of the function (in this case to only f(x)>0)

 

[daon]

However,

is it legitimate in the particular above case to use critical numbers in a function to find intervals of increase/decrease by testing a number on each side of the critical numbers to see if it is positive/negative? (Positive meaning increasing, negative decreasing on that interval by the I/D test)

It makes sense to me that it should be, since the increasing/decreasing of the function should be unchanged at the interval in question even if you shrink the domain of the function (in this case to only f(x)>0)

Since this particular differentiable function has no horizontal tangent on (0,oo) you know it will never change in that interval. In general, testing points between critical numbers in the derivative will tell you whether it is increasing or decreasing in that interval as you have stated. Just be careful not to miss any critical values...