Check Full Chapter Explained - Continuity and Differentiability - Continuity and Differentiability Class 12



Last updated at Jan. 3, 2020 by Teachoo
Check Full Chapter Explained - Continuity and Differentiability - Continuity and Differentiability Class 12
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 ๐ฅ=1 and ๐ฅ= 2. (๐๐๐)โฌ(hโ0) (๐(๐ฅ) โ ๐(๐ฅ โ โ))/โ = (๐๐๐)โฌ(hโ0) (๐(1) โ ๐(1 โ โ))/โ = (๐๐๐)โฌ(hโ0) ([1] โ [(1 โ โ)])/โ = (๐๐๐)โฌ(hโ0) (1 โ 0)/โ = (๐๐๐)โฌ(hโ0) 1/โ = 1/0 = Not defined (๐๐๐)โฌ(hโ0) (๐(๐ฅ + โ) โ ๐(๐ฅ))/โ = (๐๐๐)โฌ(hโ0) (๐(1 + โ) โ ๐(1))/โ = (๐๐๐)โฌ(hโ0) ([(1 + โ)] โ [1])/โ = (๐๐๐)โฌ(hโ0) (1 โ 1)/โ = (๐๐๐)โฌ(hโ0) 0/โ = (๐๐๐)โฌ(hโ0) 0 = 0 For greatest integer function [1 + h] = 1 [1 โ h] = 0 [1] = 1 (๐๐๐)โฌ(hโ0) (๐(๐ฅ) โ ๐(๐ฅ โ โ))/โ = (๐๐๐)โฌ(hโ0) (๐(2) โ ๐(2 โ โ))/โ = (๐๐๐)โฌ(hโ0) ([2] โ [(2 โ โ)])/โ = (๐๐๐)โฌ(hโ0) (2 โ 1)/โ = (๐๐๐)โฌ(hโ0) 1/โ = 1/0 = Not defined (๐๐๐)โฌ(hโ0) (๐(๐ฅ + โ) โ ๐(๐ฅ))/โ = (๐๐๐)โฌ(hโ0) (๐(2 + โ) โ ๐(2))/โ = (๐๐๐)โฌ(hโ0) ([(2 + โ)] โ [2])/โ = (๐๐๐)โฌ(hโ0) (2 โ 2)/โ = (๐๐๐)โฌ(hโ0) 0/โ = (๐๐๐)โฌ(hโ0) 0 = 0 For greatest integer function [2 + h] = 2 [2 โ h] = 1 [2] = 2 Since LHD โ RHD โด f(x) is not differentiable at x = 1 Hence proved (๐๐๐)โฌ(hโ0) (๐(๐ฅ) โ ๐(๐ฅ โ โ))/โ = (๐๐๐)โฌ(hโ0) (๐(2) โ ๐(2 โ โ))/โ = (๐๐๐)โฌ(hโ0) ([2] โ [(2 โ โ)])/โ = (๐๐๐)โฌ(hโ0) (2 โ 1)/โ = (๐๐๐)โฌ(hโ0) 1/โ = 1/0 = Not defined (๐๐๐)โฌ(hโ0) (๐(๐ฅ + โ) โ ๐(๐ฅ))/โ = (๐๐๐)โฌ(hโ0) (๐(2 + โ) โ ๐(2))/โ = (๐๐๐)โฌ(hโ0) ([(2 + โ)] โ [2])/โ = (๐๐๐)โฌ(hโ0) (2 โ 2)/โ = (๐๐๐)โฌ(hโ0) 0/โ = (๐๐๐)โฌ(hโ0) 0 = 0 For greatest integer function [2 + h] = 2 [2 โ h] = 1 [2] = 2 For greatest integer function [2 + h] = 2 [2 โ h] = 1 [2] = 2 For greatest integer function [2 + h] = 2 [2 โ h] = 1 [2] = 2 Since LHD โ RHD โด f(x) is not differentiable at x = 2 Hence proved
About the Author