Question

Find the interval in which the function () = log( − 1) is increasing?​

Answer 1

Step-by-step explanation:

Given: f(x)=log(1+x)−

1+x

x

To find: intervals at which f(x) is increasing or decreasing.

solution: f(x)=log(1+x)−

1+x

x

f

′

(x)=

1+x

1

−[

(1+x)

2

(1)(1+x)−x(0+1)

=

1+x

1

−[

(1+x)

2

1+x−x

]

=

1+x

1

−

(1+x)

2

1

=

(1+x)

2

1+x−1

=

(1+x)

2

x

For f(x) to be increasing, f

′

(x)>0

(1+x)

2

x

>0

x<0 [∵(1+x)

2

>0, Domain: (−1,∞)]

x∈(−1,0) for f(x) to be decreasing

∴f(x) is increasing on (0,∞)

decreasing on (−1,0).