Ex 5.2, 9 - Prove that f(x) = |x - 1| is not differentiable

Ex 5.2, 9 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Ex 5.2, 9 - Chapter 5 Class 12 Continuity and Differentiability - Part 3

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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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.