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

  1. Chapter 6 Class 12 Application of Derivatives (Term 1)
  2. Serial order wise

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

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.