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

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Answer 1 · 2017-09-08T13:03:29

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

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]}