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

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


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

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.