Check sibling questions

Ex 6.2,7 - Ex 6.2

Ex 6.2,7 - Chapter 6 Class 12 Application of Derivatives - Part 2
Ex 6.2,7 - Chapter 6 Class 12 Application of Derivatives - Part 3
Ex 6.2,7 - Chapter 6 Class 12 Application of Derivatives - Part 4

Solve all your doubts with Teachoo Black (new monthly pack available now!)


Transcript

Ex 6.2, 7 Show that ๐‘ฆ = log(1 + ๐‘ฅ) โ€“ 2๐‘ฅ/(2 + ๐‘ฅ) , ๐‘ฅ > โ€“ 1 , is an increasing function of ๐‘ฅ throughout its domain.Given ๐‘ฆ = log (1+๐‘ฅ) โ€“ 2๐‘ฅ/(2 + ๐‘ฅ) , ๐‘ฅ > โ€“1 We need to show that y is strictly increasing function for ๐‘ฅ > โ€“1 i.e. we need to show that (๐’…๐’š )/๐’…๐’™ > 0 for ๐’™ > โ€“1 Finding ๐’…๐’š/๐’…๐’™ ๐‘ฆ = log (1+๐‘ฅ) โ€“ (2๐‘ฅ )/(2 + ๐‘ฅ) (๐‘‘๐‘ฆ )/๐‘‘๐‘ฅ = ๐‘‘(logโกใ€–(1 + ๐‘ฅ) โˆ’ 2๐‘ฅ/(2 + ๐‘ฅ)ใ€— )/๐‘‘๐‘ฅ (๐‘‘๐‘ฆ )/๐‘‘๐‘ฅ = ๐‘‘(๐‘™๐‘œ๐‘”(1 + ๐‘ฅ))/๐‘‘๐‘ฅ โ€“ ๐‘‘/๐‘‘๐‘ฅ (๐Ÿ๐’™/(๐Ÿ+๐’™)) (๐‘‘๐‘ฆ )/๐‘‘๐‘ฅ = 1/(1 + ๐‘ฅ) . (1+๐‘ฅ)โ€™ โ€“ [((๐Ÿ๐’™)^โ€ฒ (๐Ÿ + ๐’™) โˆ’ใ€– (๐Ÿ + ๐’™)ใ€—^โ€ฒ ๐Ÿ๐’™)/(๐Ÿ + ๐’™)๐Ÿ] (๐‘‘๐‘ฆ )/๐‘‘๐‘ฅ = 1/(1 + ๐‘ฅ) . (0+1) โ€“ [(2(2 + ๐‘ฅ) โˆ’ (0 + 1)2๐‘ฅ)/(2 + ๐‘ฅ)2] (๐‘‘๐‘ฆ )/๐‘‘๐‘ฅ = 1/(1 + ๐‘ฅ) โ€“[(4 + 2๐‘ฅ โˆ’ 2๐‘ฅ)/(2 + ๐‘ฅ)2] (๐‘‘๐‘ฆ )/๐‘‘๐‘ฅ = 1/(1 + ๐‘ฅ) โ€“ [4/(2 + ๐‘ฅ)2] (๐‘‘๐‘ฆ )/๐‘‘๐‘ฅ = ((2 + ๐‘ฅ)2 โˆ’ 4(1 + ๐‘ฅ))/(1 + ๐‘ฅ)(2 + ๐‘ฅ)2 (๐‘‘๐‘ฆ )/๐‘‘๐‘ฅ = ((2)2 + (๐‘ฅ)2 + 2(2)(๐‘ฅ) โˆ’ 4 โˆ’ 4๐‘ฅ)/(1 + ๐‘ฅ)(2 + ๐‘ฅ)2 (๐‘‘๐‘ฆ )/๐‘‘๐‘ฅ = (4 + ๐‘ฅ2 + 4๐‘ฅ โˆ’ 4 โˆ’ 4๐‘ฅ)/(1 + ๐‘ฅ)(2 + ๐‘ฅ)2 (๐‘‘๐‘ฆ )/๐‘‘๐‘ฅ = ๐‘ฅ2/(1 + ๐‘ฅ)(2 + ๐‘ฅ)2 (๐‘‘๐‘ฆ )/๐‘‘๐‘ฅ = (๐‘ฅ/(2 + ๐‘ฅ))^2 1/(1 + ๐‘ฅ) Now, (๐‘‘๐‘ฆ )/๐‘‘๐‘ฅ = (๐’™/(๐Ÿ + ๐’™))^๐Ÿ 1/(1 + ๐‘ฅ) Now, finding value where ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ > 0 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ > 0 (๐‘ฅ/(2 + ๐‘ฅ))^2.(1/(1 + ๐‘ฅ)) > 0 (๐Ÿ/(๐Ÿ + ๐’™ ))>๐ŸŽ This is possible only when 1 + ๐‘ฅ > 0 i.e. ๐’™ > โ€“1 So, ๐’…๐’š/๐’…๐’™ > 0 for ๐’™ > โ€“1 Hence proved

Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.