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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise

Transcript

Ex 5.2, 10 (Introduction) Greatest Integer Function f(x) = [x] than or equal to x. Greatest Integer Function [x] Going by same Concept Ex 5.2, 10 Prove that the greatest integer function defined by f (x) = [x], 0 < x < 3 is not differentiable at ๐‘ฅ=1 and ๐‘ฅ= 2. (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(๐‘ฅ) โˆ’ ๐‘“(๐‘ฅ โˆ’ โ„Ž))/โ„Ž = (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’ โ„Ž)])/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (1 โˆ’ 0)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 1/โ„Ž = 1/0 = Not defined (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(๐‘ฅ + โ„Ž) โˆ’ ๐‘“(๐‘ฅ))/โ„Ž = (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])/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (1 โˆ’ 1)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 0/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 0 = 0 For greatest integer function [1 + h] = 1 [1 โˆ’ h] = 0 [1] = 1 (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(๐‘ฅ) โˆ’ ๐‘“(๐‘ฅ โˆ’ โ„Ž))/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(2) โˆ’ ๐‘“(2 โˆ’ โ„Ž))/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) ([2] โˆ’ [(2 โˆ’ โ„Ž)])/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (2 โˆ’ 1)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 1/โ„Ž = 1/0 = Not defined (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(๐‘ฅ + โ„Ž) โˆ’ ๐‘“(๐‘ฅ))/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(2 + โ„Ž) โˆ’ ๐‘“(2))/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) ([(2 + โ„Ž)] โˆ’ [2])/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (2 โˆ’ 2)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 0/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 0 = 0 For greatest integer function [2 + h] = 2 [2 โˆ’ h] = 1 [2] = 2 Since LHD โ‰  RHD โˆด f(x) is not differentiable at x = 1 Hence proved (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(๐‘ฅ) โˆ’ ๐‘“(๐‘ฅ โˆ’ โ„Ž))/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(2) โˆ’ ๐‘“(2 โˆ’ โ„Ž))/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) ([2] โˆ’ [(2 โˆ’ โ„Ž)])/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (2 โˆ’ 1)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 1/โ„Ž = 1/0 = Not defined (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(๐‘ฅ + โ„Ž) โˆ’ ๐‘“(๐‘ฅ))/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(2 + โ„Ž) โˆ’ ๐‘“(2))/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) ([(2 + โ„Ž)] โˆ’ [2])/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (2 โˆ’ 2)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 0/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 0 = 0 For greatest integer function [2 + h] = 2 [2 โˆ’ h] = 1 [2] = 2 For greatest integer function [2 + h] = 2 [2 โˆ’ h] = 1 [2] = 2 For greatest integer function [2 + h] = 2 [2 โˆ’ h] = 1 [2] = 2 Since LHD โ‰  RHD โˆด f(x) is not differentiable at x = 2 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 9 years. He provides courses for Maths and Science at Teachoo.