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, 8 Find the values of ๐‘ฅ for which y = [๐‘ฅ(๐‘ฅ โ€“ 2)]2 is an increasing function ๐‘ฆ = [๐‘ฅ(๐‘ฅโˆ’2)]^2 Finding ๐’…๐’š/๐’…๐’™ ๐‘ฆ =[๐‘ฅ(๐‘ฅโˆ’2)]^2 ๐‘ฆ =[๐‘ฅ^2โˆ’2๐‘ฅ]^2 ๐‘ฆ =(๐‘ฅ)^4+(2๐‘ฅ)^2โˆ’2(๐‘ฅ^2 )(2๐‘ฅ) ๐‘ฆ = ๐‘ฅ^4+4๐‘ฅ^2โˆ’4๐‘ฅ^3 Diff. w.r.t ๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=๐‘‘(๐‘ฅ^4 + 4๐‘ฅ^2 โˆ’ 4๐‘ฅ^3 )/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=4๐‘ฅ^3+8๐‘ฅโˆ’12๐‘ฅ^2 ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=4๐‘ฅ(๐‘ฅ^2+2โˆ’3๐‘ฅ) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=4๐‘ฅ(๐‘ฅ^2โˆ’3๐‘ฅ+2) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=4๐‘ฅ(๐‘ฅ^2โˆ’2๐‘ฅโˆ’๐‘ฅ+2) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=4๐‘ฅ(๐‘ฅ(๐‘ฅโˆ’2)โˆ’1(๐‘ฅโˆ’2)) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=4๐‘ฅ((๐‘ฅโˆ’1)(๐‘ฅโˆ’2)) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ=4๐‘ฅ(๐‘ฅโˆ’1)(๐‘ฅโˆ’2) Putting ๐’…๐’š/๐’…๐’™=๐ŸŽ 4๐‘ฅ(๐‘ฅโˆ’1)(๐‘ฅโˆ’2)=0 So, ๐‘ฅ=0 , ๐‘ฅ=1 & ๐‘ฅ=2 Plotting points on real line The points ๐‘ฅ = 0 , 1 and 2 divide the real line into 4 disjoint intervals i.e. (โˆ’โˆž,0) , (0 , 1) , (1 , 2) & (2 ,โˆž) Thus the function y = [๐‘ฅ(๐‘ฅโˆ’1)^2 ] is strictly increasing for 0 <๐‘ฅ<1 and ๐‘ฅ>2

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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.