Check sibling questions

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 17 The function f (x) = 𝑒^(|π‘₯|) 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(π‘₯) = 𝑒^(|π‘₯|) We need to check continuity and differentiability of f(π‘₯) Continuity of f(𝒙) Let π’ˆ(𝒙)=𝒆^𝒙 & 𝒉(𝒙)=|𝒙| Then, π’ˆπ’π’‰(𝒙)=𝑔(β„Ž(π‘₯)) =𝑔(|π‘₯|) =𝑒^|π‘₯| =𝒇(𝒙) ∴ 𝑓(π‘₯)=π‘”π‘œβ„Ž(π‘₯) We know that, 𝒉(𝒙)=|𝒙| is continuous as it is modulus function π’ˆ(𝒙)=𝑒^π‘₯ is continuous as it is an exponential function Hence, g(π‘₯) & h(π‘₯) both are continuous And If two functions g(π‘₯) & h(π‘₯) are continuous then their composition π‘”π‘œβ„Ž(π‘₯) is also continuous ∴ 𝒇(𝒙) is continuous Differentiability of 𝒇(𝒙) 𝑓(π‘₯)=𝑒^(|π‘₯|) 𝑓(π‘₯)={β– 8(𝑒^π‘₯, π‘₯β‰₯0@𝑒^(βˆ’π‘₯), π‘₯<0)─ Now, 𝑓(π‘₯) is differentiable at π‘₯=0, if LHD = RHD (π’π’Šπ’Ž )┬(π‘β†’πŸŽ) (𝒇(𝒙) βˆ’ 𝒇(𝒙 βˆ’ 𝒉))/𝒉 = (π‘™π‘–π‘š)┬(hβ†’0) (𝑓(0) βˆ’ 𝑓(0 βˆ’ β„Ž))/β„Ž = (π‘™π‘–π‘š)┬(hβ†’0) (𝑒^(|0|)βˆ’ 𝑒^(|0 βˆ’β„Ž|))/β„Ž = (π‘™π‘–π‘š)┬(hβ†’0) (𝑒^(|0|)βˆ’ 𝑒^(| βˆ’β„Ž|))/β„Ž = (π‘™π‘–π‘š)┬(hβ†’0) (𝑒^0 βˆ’ 𝑒^β„Ž)/β„Ž = (π‘™π‘–π‘š)┬(hβ†’0) (1 βˆ’ 𝑒^β„Ž)/β„Ž = (π‘™π‘–π‘š)┬(hβ†’0) (βˆ’(𝑒^β„Ž βˆ’ 1))/β„Ž Using (π‘™π‘–π‘š)┬(xβ†’0) (𝑒^π‘₯ βˆ’ 1)/π‘₯=1 = (π‘™π‘–π‘š)┬(hβ†’0) βˆ’1 = βˆ’1 (π’π’Šπ’Ž )┬(π‘β†’πŸŽ) (𝒇(𝒙 + 𝒉) βˆ’ 𝒇(𝒙 ))/𝒉 = (π‘™π‘–π‘š)┬(hβ†’0) (𝑓(0 + β„Ž) βˆ’ 𝑓(0))/β„Ž = (π‘™π‘–π‘š)┬(hβ†’0) (𝑒^(|0 + β„Ž|) βˆ’π‘’^(|0|))/β„Ž = (π‘™π‘–π‘š)┬(hβ†’0) (𝑒^(|β„Ž|) βˆ’π‘’^0)/β„Ž = (π‘™π‘–π‘š)┬(hβ†’0) (𝑒^β„Ž βˆ’ 1)/β„Ž Using (π‘™π‘–π‘š)┬(xβ†’0) (𝑒^π‘₯ βˆ’ 1)/π‘₯=1 = (π‘™π‘–π‘š)┬(hβ†’0) 1 = 𝟏 Since, LHD β‰  RHD ∴ 𝑓(π‘₯) is not differentiable at π‘₯=0 Thus, 𝑓(π‘₯) continuous everywhere but not differentiable at x = 0 So, the correct answer is (A)

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.