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



Last updated at Jan. 7, 2020 by Teachoo
Check Full Chapter Explained - Continuity and Differentiability - Application of Derivatives (AOD) Class 12
Transcript
Ex 6.2, 8 Find the values of ๐ฅ for which y = [๐ฅ(๐ฅ โ 2)]2 is an increasing function ๐ฆ = [๐ฅ(๐ฅโ2)]^2 Finding ๐ ๐/๐ ๐ ๐ฆ =[๐ฅ(๐ฅโ2)]^2 ๐ฆ =[๐ฅ^2โ2๐ฅ]^2 ๐ฆ =(๐ฅ)^4+(2๐ฅ)^2โ2(๐ฅ^2 )(2๐ฅ) ๐ฆ = ๐ฅ^4+4๐ฅ^2โ4๐ฅ^3 Diff. w.r.t ๐ฅ ๐๐ฆ/๐๐ฅ=๐(๐ฅ^4 + 4๐ฅ^2 โ 4๐ฅ^3 )/๐๐ฅ ๐๐ฆ/๐๐ฅ=4๐ฅ^3+8๐ฅโ12๐ฅ^2 ๐๐ฆ/๐๐ฅ=4๐ฅ(๐ฅ^2+2โ3๐ฅ) ๐๐ฆ/๐๐ฅ=4๐ฅ(๐ฅ^2โ3๐ฅ+2) ๐๐ฆ/๐๐ฅ=4๐ฅ(๐ฅ^2โ2๐ฅโ๐ฅ+2) ๐๐ฆ/๐๐ฅ=4๐ฅ(๐ฅ(๐ฅโ2)โ1(๐ฅโ2)) ๐๐ฆ/๐๐ฅ=4๐ฅ((๐ฅโ1)(๐ฅโ2)) ๐๐ฆ/๐๐ฅ=4๐ฅ(๐ฅโ1)(๐ฅโ2) Putting ๐ ๐/๐ ๐=๐ 4๐ฅ(๐ฅโ1)(๐ฅโ2)=0 So, ๐ฅ=0 , ๐ฅ=1 & ๐ฅ=2 Plotting points on real line The points ๐ฅ = 0 , 1 and 2 divide the real line into 4 disjoint intervals i.e. (โโ,0) , (0 , 1) , (1 , 2) & (2 ,โ) Thus the function y = [๐ฅ(๐ฅโ1)^2 ] is strictly increasing for 0 <๐ฅ<1 and ๐ฅ>2
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