UPSC NDA Math Model Paper 6

Given: f(x)=x+

1

x

As we know that for a strictly increasing function f'(x) > 0 for all x ∈ (a, b)
Let's find f'(x)
As we know that,

d

dx

{f(x)±g(x)}=

d{f(x)}

dx

±

d{g(x)}

dx

⇒f′(x)=1−

1

x2

=

x2−1

x2

⇒

x2−1

x2

>0 .....(∵f(x) is a strictly increasing function)
⇒x2−1>0
⇒ (x - 1) ⋅ (x + 1) > 0
⇒ x < - 1 or x > 1
⇒ x ∈ (- ∞, - 1) ∪ (1, ∞)
Hence, the interval in which the given function f(x) is strictly increasing is: (- ∞, - 1) ∪ (1, ∞)