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Show that Y = `Log(1+X) - (2x)/(2+X), X> - 1`, Is an Increasing Function Of X Throughout Its Domain. - CBSE (Commerce) Class 12 - Mathematics

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Question

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

Solution

We have,

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

∴dydx=11+x-(2+x)(2)-2x(1)(2+x)2=11+x-4(2+x)2=x2(1+x)(2+x)2
Now, dydx=0
⇒x2(1+x)(2+x)2=0⇒x2=0      [(2+x)≠0 as x>-1]⇒x=0

Since > −1, point = 0 divides the domain (−1, ∞) in two disjoint intervals i.e., −1 < x < 0 and x > 0.

When −1 < < 0, we have:
x<0⇒x2>0x>-1⇒(2+x)>0⇒(2+x2)>0
∴ y'=x2(1+x)(2+x)2>0

Also, when x > 0:
x>0⇒x2>0, (2+x)2>0
∴ y'=x2(1+x)(2+x)2>0

Hence, function f is increasing throughout this domain

  Is there an error in this question or solution?

APPEARS IN

 NCERT Solution for Mathematics Textbook for Class 12 (2018 to Current)
Chapter 6: Application of Derivatives
Q: 7 | Page no. 205
Solution Show that Y = `Log(1+X) - (2x)/(2+X), X> - 1`, Is an Increasing Function Of X Throughout Its Domain. Concept: Increasing and Decreasing Functions.
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