Show that y = log (1 + x) - 2x/(2 + x), x > - 1 is an increasing function of x throughout its domain

Solution:

It is given that

y = log (1 + x) - 2x/(2 + x)

dy/dx

= 1/(1 + x) - [(2 + x)(2) - 2x (1)]/(2 + x)²

= 1/(1 + x) - 4/(2 + x)²

= x²/(1 + x)(2 + x)²

Now,

dy/dx = 0

Hence,

x²/(1 + x)(2 + x)² = 0

⇒ x² = 0

⇒ x = 0

Since, x > - 1, x = 0 divides domain (- 1, ∞) in two intervals - 1 < x < 0 and x > 0

When, - 1 < x < 0

Then,

x < 0

⇒ x² > 0

x > - 1

⇒ (2 + x) > 0

⇒ (2 + x)² > 0

Hence,

y = x²/(2 + x)² > 0

When, x > 0

Then,

x > 0

⇒ x² > 0

⇒ (2 + x)² > 0

Hence,

y = x² / (2 + x)² > 0

Thus, f is increasing throughout the domain

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.2 Question 7

Show that y = log (1 + x) - 2x/(2 + x), x > - 1 is an increasing function of x throughout its domain.

Summary:

Hence we have concluded that y = log (1 + x) - 2x/(2 + x), x > - 1 is an increasing function of x throughout its domain