Check Full Chapter Explained - Continuity and Differentiability - Application of Derivatives (AOD) Class 12

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

Transcript

Ex 6.2, 19 The interval in which ๐‘ฆ = ๐‘ฅ2 ๐‘’^(โ€“๐‘ฅ) is increasing is (A) (โ€“ โˆž, โˆž) (B) (โ€“ 2, 0) (C) (2, โˆž) (D) (0, 2)Let f(๐‘ฅ) = ๐‘ฅ^2 ๐‘’^(โˆ’๐‘ฅ) Finding fโ€™(๐’™) fโ€™(๐‘ฅ) = (๐‘ฅ^2 ๐‘’^(โˆ’๐‘ฅ) )โ€ฒ Using product rule fโ€™(๐‘ฅ) = (๐‘ฅ2)โ€ฒ ๐‘’^(โˆ’๐‘ฅ) + (๐‘’^(โˆ’๐‘ฅ) )โ€™ (๐‘ฅ2) fโ€™(๐‘ฅ) = (2๐‘ฅ) ๐‘’^(โˆ’๐‘ฅ) + (ใ€–โˆ’๐‘’ใ€—^(โˆ’๐‘ฅ) ) (๐‘ฅ2) fโ€™(๐‘ฅ) = 2๐‘ฅ ๐‘’^(โˆ’๐‘ฅ)โˆ’๐‘’^(โˆ’๐‘ฅ) ๐‘ฅ2 fโ€™(๐’™) = ๐’™ e โ€“๐’™ (๐Ÿโˆ’๐’™) Putting fโ€™(๐’™)=๐ŸŽ ๐’™ e โ€“๐’™ (๐Ÿโˆ’๐’™)=๐ŸŽ ๐‘ฅ (2โˆ’๐‘ฅ) = 0 So, ๐‘ฅ=0 & ๐‘ฅ = 2 Plotting points on real line (As e โ€“๐‘ฅ is always positive for all ๐‘ฅ โˆˆ R) Hence, f(๐‘ฅ) is strictly increasing on (0 , 2) Therefore, correct answer is (๐ƒ)

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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.