`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`

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

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

`0=(1-x^2)/(x^2+1)^2`

`0=1 - x^2`

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

`x=-1`

`x=1`

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.

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

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`...

(The entire section contains 623 words.)

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