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

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Transcript

Question 15 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)

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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.