The function f (x) = e |x| is

(A) continuous everywhere but not differentiable at x = 0

(B) continuous and differentiable everywhere

(C) not continuous at x = 0

(D) none of these.

This question is similar to Ex 5.1, 32 - Chapter 5 Class 12 and Ex 5.2, 9 - Chapter 5 Class 12 - Continuity and Differentiability

Slide76.JPG

Slide77.JPG
Slide78.JPG
Slide79.JPG
Slide80.JPG

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

Transcript

Question 17 The function f (x) = ๐‘’^(|๐‘ฅ|) is (A) continuous everywhere but not differentiable at x = 0 (B) continuous and differentiable everywhere (C) not continuous at x = 0 (D) none of these. f(๐‘ฅ) = ๐‘’^(|๐‘ฅ|) We need to check continuity and differentiability of f(๐‘ฅ) Continuity of f(๐’™) Let ๐’ˆ(๐’™)=๐’†^๐’™ & ๐’‰(๐’™)=|๐’™| Then, ๐’ˆ๐’๐’‰(๐’™)=๐‘”(โ„Ž(๐‘ฅ)) =๐‘”(|๐‘ฅ|) =๐‘’^|๐‘ฅ| =๐’‡(๐’™) โˆด ๐‘“(๐‘ฅ)=๐‘”๐‘œโ„Ž(๐‘ฅ) We know that, ๐’‰(๐’™)=|๐’™| is continuous as it is modulus function ๐’ˆ(๐’™)=๐‘’^๐‘ฅ is continuous as it is an exponential function Hence, g(๐‘ฅ) & h(๐‘ฅ) both are continuous And If two functions g(๐‘ฅ) & h(๐‘ฅ) are continuous then their composition ๐‘”๐‘œโ„Ž(๐‘ฅ) is also continuous โˆด ๐’‡(๐’™) is continuous Differentiability of ๐’‡(๐’™) ๐‘“(๐‘ฅ)=๐‘’^(|๐‘ฅ|) ๐‘“(๐‘ฅ)={โ– 8(๐‘’^๐‘ฅ, ๐‘ฅโ‰ฅ0@๐‘’^(โˆ’๐‘ฅ), ๐‘ฅ<0)โ”ค Now, ๐‘“(๐‘ฅ) is differentiable at ๐‘ฅ=0, 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‘“(0) โˆ’ ๐‘“(0 โˆ’ โ„Ž))/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘’^(|0|)โˆ’ ๐‘’^(|0 โˆ’โ„Ž|))/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘’^(|0|)โˆ’ ๐‘’^(| โˆ’โ„Ž|))/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘’^0 โˆ’ ๐‘’^โ„Ž)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (1 โˆ’ ๐‘’^โ„Ž)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (โˆ’(๐‘’^โ„Ž โˆ’ 1))/โ„Ž Using (๐‘™๐‘–๐‘š)โ”ฌ(xโ†’0) (๐‘’^๐‘ฅ โˆ’ 1)/๐‘ฅ=1 = (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‘“(0 + โ„Ž) โˆ’ ๐‘“(0))/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘’^(|0 + โ„Ž|) โˆ’๐‘’^(|0|))/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘’^(|โ„Ž|) โˆ’๐‘’^0)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘’^โ„Ž โˆ’ 1)/โ„Ž Using (๐‘™๐‘–๐‘š)โ”ฌ(xโ†’0) (๐‘’^๐‘ฅ โˆ’ 1)/๐‘ฅ=1 = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 1 = ๐Ÿ Since, LHD โ‰  RHD โˆด ๐‘“(๐‘ฅ) is not differentiable at ๐‘ฅ=0 Thus, ๐‘“(๐‘ฅ) continuous everywhere but not differentiable at x = 0 So, the correct answer is (A)

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.