Let I be any interval disjoint from [- 1, 1]. Prove that the function f given by f (x) = x + 1/x is increasing on I

Solution:

Increasing functions are those functions that increase monotonically within a particular domain,

and decreasing functions are those which decrease monotonically within a particular domain.

We have,

f (x) = x + 1/x

Therefore,

f' (x) = 1 - 1/x²

Now,

f' (x) = 0

⇒ 1 - 1/x² = 0

⇒ x² = 1

⇒ x = ± 1

The points x = 1 and x = - 1 divide the real line intervals (- ∞, 1), (- 1, 1) and (1, ∞)

In interval (- 1, 1) ,

- 1 < x < 1

⇒ x² < 1

⇒ 1 < 1/x² x ≠ 0

⇒ 1 - 1/x² < 0 x ≠ 0

Therefore,

 f' (x) = 1 - 1/x² < 0 on (- 1, 1) ~ {0}

Hence, f is strictly decreasing on (- 1, 1) ~ {0}

Now,

in interval (- ∞, - 1) and (1, ∞), x < - 1 or 1 < x

⇒ x² > 1

⇒ 1 > 1/x²

⇒ 1 - 1/x² > 0

Therefore,

f' (x) = 1 - 1/x² > 0 on (- ∞, - 1) and (1, ∞)

Hence,

f is strictly increasing on (- ∞, - 1) and (1, ∞)

Thus,

f is strictly increasing in I in [- 1, 1]

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.2 Question 15

Let I be any interval disjoint from [- 1, 1] . Prove that the function f given by f (x) = x + 1/x is increasing on I

Summary:

f is strictly increasing on (- ∞, - 1) and (1, ∞).Hence we have proved that the function f is given by f (x) = x + 1/x is increasing on I