UPSC NDA Math Model Paper 7

Given : f(x)=log(1+x)−

x

(1+x)

Here we have to find the interval in which f(x) is increasing.
Let's first calculate f'(x)
As we know that

d

dx

{f(x)±g(x)}=

d{f(x)}

dx

±

d{g(x)}

dx

,

d(log‌x)

dx

=

1

x

,for x>0 and

d

dx

[

f(x)

g(x)

] =

g(x). d(f(x)) dx −f(x). d(g(x)) dx

[g(x)]2

⇒f′(x)={

1

1+x

−

(1+x).1−x.1

(1+x2)

} =

x

(1+x)2

As we know that for an increasing function say f(x) we have f′(x)≥0
⇒x≥0.........(∵(1+x)2≥0)
Hence, the given function is increasing for the interval [0, ∞)