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  1. Chapter 6 Class 12 Application of Derivatives
  2. Concept 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 fโ€™(๐‘ฅ) = (๐‘ฅ2)โ€ฒ e โ€“๐‘ฅ + (๐‘’^(โˆ’๐‘ฅ) )โ€™ (๐‘ฅ2) fโ€™(๐‘ฅ) = (2๐‘ฅ) e โ€“๐‘ฅ + (ใ€–โˆ’๐‘’ใ€—^(โˆ’๐‘ฅ) ) (๐‘ฅ2) fโ€™(๐‘ฅ) = 2๐‘ฅ e-๐‘ฅ โ€“ e-๐‘ฅ ๐‘ฅ2 fโ€™(๐‘ฅ) = ๐‘ฅ e โ€“๐‘ฅ (2โˆ’๐‘ฅ) Putting fโ€™(๐’™)=๐ŸŽ ๐‘ฅ e-๐‘ฅ (2โˆ’๐‘ฅ) = 0 ๐‘ฅ (2โˆ’๐‘ฅ) = 0 So, ๐‘ฅ=0 & ๐‘ฅ = 2 Plotting points on real line The points ๐‘ฅ = 0 & 2 divide the real line into 3 disjoint intervals i.e (โˆ’โˆž , 0) , (0 , 2) & (2 , โˆž) (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 9 years. He provides courses for Maths and Science at Teachoo.