Question

How do you determine the intervals for which the function is increasing or decreasing given `f(x)=1/(x+1)-4`?

Answer 1

The intervals of decreasing are `(-oo,-1) uu (-1,+oo)`

Explanation:

We need

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

`f(x)=1/(1+x)-4`

The domain of `f(x)` is `D_f(x)=RR-{-1}`

Therefore, the derivative of `f(x)` is

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

`AA x in D_f(x), f'(x)<0`

There are no critical values, and the variation table is

`color(white)(aaaa)` `Interval` `color(white)(aaaa)` `(-oo,-1)` `color(white)(aaaa)` `(-1,+oo)`

`color(white)(aaaa)` `sign f'(x)` `color(white)(aaaaaaaa)` `-` `color(white)(aaaaaaaaaaa)` `-`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaaaaaaaa)` `↘` `color(white)(aaaaaaaaaaa)` `↘`

You can also, calculate the second derivative and determine the concavity.

graph{1/(1+x)-4 [-10, 10, -5, 5]}