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, 15 Let I be any interval disjoint from [โ1, 1]. Prove that the function f given by ๐(๐ฅ) = ๐ฅ + 1/๐ฅ is strictly increasing on I. I is any interval disjoint from [โ1, 1] Hence I = (โโ, โ1)โช(1, โ) f(๐ฅ) = ๐ฅ + 1/๐ฅ We need to show f(๐ฅ) is strictly increasing on I i.e. we need to show fโ(๐ฅ) > 0 for ๐ฅ โ(โโ, โ1)โช(1, โ) Finding fโ(๐) f(๐ฅ) = ๐ฅ + 1/๐ฅ fโ(๐ฅ) = 1 โ 1/๐ฅ2 fโ(๐ฅ) = (๐ฅ2 โ 1)/๐ฅ2 Putting fโ(๐) = 0 (๐ฅ2 โ 1)/๐ฅ2 = 0 ๐ฅ2โ1 = 0 (๐ฅโ1)(๐ฅ+1)=0 So, ๐ฅ=1 & ๐ฅ=โ1 Plotting point on real line The point ๐ฅ = โ1 , 1 into three disjoint intervals i.e. (โโ , โ1) (โ1 , 1) & (1 , โ) โด f(x) is strictly increasing on (โโ , โ๐) & (๐ , โ) Hence proved
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