`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`
`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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