This question is similar to Chapter 5 Class 12 Continuity and Differentiability - Ex 5.1

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Question 19 Statement A Assertion (A): Consider the function defined as 𝑓(π‘₯)=|π‘₯|+|π‘₯βˆ’1|,π‘₯βˆˆπ‘…. Then 𝑓(π‘₯) is not differentiable at 𝒙=𝟎 and 𝒙=𝟏. Statement R Reason (R): Suppose 𝑓 be defined and continuous on (π‘Ž,𝑏) and π‘βˆˆ(π‘Ž,𝑏), then 𝑓(π‘₯) is not differentiable at π‘₯=𝑐 if lim_(β„Žβ†’0^βˆ’ )β€Š(𝑓(𝑐+β„Ž)βˆ’π‘“(𝑐))/β„Žβ‰ lim_(β„Žβ†’0^+ )β€Š(𝑓(𝑐 + β„Ž)βˆ’π‘“(𝑐))/β„Ž. (A) Both (A) and (R) are true and (R) is the correct explanation of (A). (B) Both (A) and ( 𝑅 ) are true but (𝑅) is not the correct explanation of (𝐴). (C) (A) is true but (R) is false. (D) (A) is false but (𝑅) is true.Checking Assertion Assertion (A): Consider the function defined as 𝑓(π‘₯)=|π‘₯|+|π‘₯βˆ’1|,π‘₯βˆˆπ‘…. Then 𝑓(π‘₯) is not differentiable at 𝒙=𝟎 and 𝒙=𝟏. Given 𝑓(π‘₯)= |π‘₯|+ |π‘₯βˆ’1|. Here, we have 2 critical points x = 0 and x βˆ’ 1 = 0 i.e. x = 0, and x = 1 Let’s draw its graph From graph, we can see that At x = 0 and x = 1, It is an edge point And, the function is not differentiable at edge points. Since at x = 0, and x = 1 is not differentiable ∴ Assertion is true Checking Reason Reason (R): Suppose 𝑓 be defined and continuous on (π‘Ž,𝑏) and π‘βˆˆ(π‘Ž,𝑏), then 𝑓(π‘₯) is not differentiable at π‘₯=𝑐 if lim_(β„Žβ†’0^βˆ’ )β€Š(𝑓(𝑐+β„Ž)βˆ’π‘“(𝑐))/β„Žβ‰ lim_(β„Žβ†’0^+ )β€Š(𝑓(𝑐 + β„Ž)βˆ’π‘“(𝑐))/β„Ž.. Here, reasoning is describing the differentiability test lim_(β„Žβ†’0^βˆ’ )β€Š(𝑓(𝑐+β„Ž)βˆ’π‘“(𝑐))/β„Ž is Left hand derivative at x = c lim_(β„Žβ†’0^+ )β€Š(𝑓(𝑐 + β„Ž)βˆ’π‘“(𝑐))/β„Ž is Right hand derivative at x = c And, since the function is not differentiable at x = c, LHD β‰  RHD This is true Hence, Reason is true Is Reason a Correct explanation for Assertion? Since Derivative test can be used to check differentiability at x = 0, and x = 1 Thus, we used the concept mentioned in Reasoning to check Assertion Therefore, Reasoning is a correct explanation for Assertion So, Assertion is true Reasoning is true And, Reasoning is a correct explanation for Assertion 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 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.