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`f(x) = x/(x^2 + 1)` (a) Find the intervals on which `f` is increasing or decreasing. (b) Find the local maximum and minimum values of `f`. (c) Find the intervals of concavity and the inflection points.

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`f(x) = x/(x^2+1)`

(a) Take the derivative of the given function.

`f'(x) = ((x^2+1)(1) - (x)(2x))/(x^2+1)^2= (x^2+1-2x^2)/(x^2+1)^2`


Then, solve for the critical numbers by setting the derivative equal to zero.


`0=1 - x^2`

`0=(1 - x)(1 + x)`



So the critical numbers are x=-1 and x=1. The intervals formed by these two critical numbers are

`(-oo,-1)`     ` (-1,1)`    and   `(1,oo)` .

Then, assign a test value for each interval and plug-in them to the first derivative.


If the resulting value of f'(x) is negative,  the function is decreasing in that interval. If it is positive, the function is increasing.

For our first interval` (-oo,-1)` , let the test value be x=-2.

`f'(-2) = (1-(-2)^2)/((-2)^2+1)^2=-3/25`    (Decreasing)

For our second interval (-1,1), let the test value be x=0.

`f'(0)=(1-0^2)/(0^2+1)^2=1`      (Increasing)

And for our third interval, let the test value be x=2.

`f(2)= (1-2^2)/(2^2+1)^2=-3/25`...

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