The set of points where the functions f given by f (x) = |x – 3| cos x is differentiable is

(A) R  

(B) R − {3}

(C) (0, ∞) 

(D) None of these

This question is similar to Ex 5.2, 9 - Chapter 5 Class 12 - Continuity and Differentiability

Slide37.JPG

Slide38.JPG
Slide39.JPG

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

Transcript

Question 8 The set of points where the functions f given by f (x) = |x โ€“ 3| cos x is differentiable is (A) R (B) R โˆ’ {3} (C) (0, โˆž) (D) None of these f(x) = |๐‘ฅโˆ’3| cosโก๐‘ฅ = {โ–ˆ((๐‘ฅโˆ’3) cosโก๐‘ฅ, ๐‘ฅโˆ’3โ‰ฅ0@โˆ’(๐‘ฅโˆ’3) cosโก๐‘ฅ, ๐‘ฅโˆ’3<0)โ”ค = {โ–ˆ((๐‘ฅโˆ’3) cosโก๐‘ฅ, ๐‘ฅโ‰ฅ3@โˆ’(๐‘ฅโˆ’3) cosโก๐‘ฅ, ๐‘ฅ<3)โ”ค Now, f(x) is a differentiable at x = 3 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‘“(3) โˆ’ ๐‘“(3 โˆ’ โ„Ž))/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (|3 โˆ’ 3| cosโก3โˆ’|(3 โˆ’ โ„Ž)โˆ’3| cosโกใ€–(3 โˆ’ โ„Ž)ใ€—)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (0 โˆ’|3 โˆ’ โ„Ž โˆ’3| cosโกใ€–(3 โˆ’ โ„Ž)ใ€—)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (0 โˆ’|โˆ’โ„Ž| cosโกใ€–(3 โˆ’ โ„Ž)ใ€—)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (โˆ’โ„Ž cosโกใ€–(3 โˆ’ โ„Ž)ใ€—)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) โˆ’cosโกใ€–(3 โˆ’โ„Ž)ใ€— = โˆ’cosโกใ€–(3 โˆ’0)ใ€— = โˆ’๐’„๐’๐’”โก๐Ÿ‘ (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‘“(3+โ„Ž) โˆ’ ๐‘“(3))/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (|(3+โ„Ž) โˆ’ 3| cosโกใ€–(3+โ„Ž)ใ€—โˆ’|3 โˆ’ 3| cosโกใ€–(3)ใ€—)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (|3 + โ„Ž โˆ’3| cosโก(3 + โ„Ž)โˆ’0 )/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (| โ„Ž| cosโกใ€–(3+โ„Ž)ใ€—)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (โ„Ž cosโกใ€–(3 + โ„Ž)ใ€—)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) cosโกใ€–(3+โ„Ž)ใ€— = cosโกใ€–(3+0)ใ€— = ๐’„๐’๐’”โก๐Ÿ‘ Since LHD โ‰  RHD โˆด f(x) is not differentiable at x = 3 Hence, we can say that f(x) is differentiable on R โˆ’ {๐Ÿ‘} So, the correct answer is (B)

About the Author

Davneet Singh's photo - Teacher, Engineer, Marketer
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.