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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Concept wise

Transcript

Ex 5.2, 9 Prove that the function f given by ๐‘“ (๐‘ฅ) = | ๐‘ฅ โ€“ 1|, ๐‘ฅ โˆˆ ๐‘… is not differentiable at x = 1. f(x) = |๐‘ฅโˆ’1| = {โ–ˆ((๐‘ฅโˆ’1), ๐‘ฅโˆ’1โ‰ฅ0@โˆ’(๐‘ฅโˆ’1), ๐‘ฅโˆ’1<0)โ”ค = {โ–ˆ((๐‘ฅโˆ’1), ๐‘ฅโ‰ฅ1@โˆ’(๐‘ฅโˆ’1), ๐‘ฅ<1)โ”ค Now, f(x) is a differentiable at x = 1 if LHD = RHD (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐’™) โˆ’ ๐’‡(๐’™ โˆ’ ๐’‰))/๐’‰ = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(1) โˆ’ ๐‘“(1 โˆ’ โ„Ž))/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (|1 โˆ’ 1|โˆ’|(1 โˆ’ โ„Ž)โˆ’1|)/โ„Ž = (๐‘™ ๐‘–๐‘š)โ”ฌ(hโ†’0) (0 โˆ’|โˆ’โ„Ž|)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (0 โˆ’ โ„Ž)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (โˆ’โ„Ž)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (โˆ’1) = โˆ’1 (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐’™ + ๐’‰) โˆ’ ๐’‡(๐’™))/๐’‰ = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(1 + โ„Ž) โˆ’ ๐‘“(1))/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (|(1 + โ„Ž) โˆ’ 1|โˆ’|1 โˆ’ 1|)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (|โ„Ž| โˆ’ 0)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (โ„Ž โˆ’ 0)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) โ„Ž/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (1) = 1 Since LHD โ‰  RHD โˆด f(x) is not differentiable at x = 1 Hence proved

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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.