Question 6 - NCERT Exemplar - MCQs - Chapter 5 Class 12 Continuity and Differentiability (Term 1)

Last updated at Nov. 17, 2021 by Teachoo

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Question 6
The function f (x) = |x| + |x β 1| is
(A) continuous at x = 0 as well as at x = 1.
(B) continuous at x = 1 but not at x = 0.
(C) discontinuous at x = 0 as well as at x = 1.
(D) continuous at x = 0 but not at x = 1.
Given π(π₯)= |π₯|+|π₯β1|
Here, we have 2 critical points
i.e. π₯ = 0, and π = 1
So, our intervals will be
When πβ€π
When 0<π<π
When πβ₯π
When πβ€π
π(π₯)= |π₯|+ |π₯β1|.
Here, both will be negative
π(π₯)=(βπ₯)+(β(π₯β1))
π(π₯)=βπ₯β(π₯β1)
π(π₯)=βπ₯βπ₯+1
" " π(π)=βππ+π
When 0<π<π
π(π₯)= |π₯| +|π₯β1|.
Here, x will be positive, but (x - 1) will be negative
π(π₯)=(π₯)+(β(π₯β1))
π(π₯)=π₯β(π₯β1)
π(π₯)=π₯βπ₯+1
" " π(π)=π
When πβ₯π
π(π₯)= |π₯|+|π₯β1|.
Here, both will be positive
π(π₯)=π₯+(π₯β1)
π(π₯)=π₯+π₯β1
" " π(π)=ππβπ
Thus, our function becomes
π(π)={β(βππ+π" " ππ πβ€π@ π ππ π<π<π@ππβπ ππ πβ₯π)β€
Since, from options we need to check continuity of the function when x = 0 and x = 1
Letβs check continuity
Case 1 : When x = 0
f(x) is continuous at π₯ =0
if L.H.L = R.H.L = π(0)
if (π₯π’π¦)β¬(π±βπ^β ) π(π)=(π₯π’π¦)β¬(π±βπ^+ ) " " π(π)= π(π)
LHL at x β 0
(πππ)β¬(π±βπ^β ) f(x) = limβ¬(hβ0) f(0 β h)
= (πππ)β¬(π‘βπ) f (β h)
= limβ¬(hβ0) (β2(ββ))+1
= limβ¬(hβ0) (2β)+1
= 2(0) + 1
= 0 + 1
= 1
RHL at x β 0
(πππ)β¬(π±βγβπγ^+ ) f(x) = limβ¬(hβ0) f(0 + h)
= (πππ)β¬(π‘βπ) f(h)
= limβ¬(hβ0) 1
= 1
Case 2 : When x = 1
f(x) is continuous at π₯=1
if L.H.L = R.H.L = π(1)
if limβ¬(xβ1^β ) π(π₯)=limβ¬(xβ1^+ ) " " π(π₯)=π(1)
LHL at x β 1
limβ¬(xβ1^β ) f(x) = limβ¬(hβ0) f(1 β h)
= limβ¬(hβ0) 1
= 1
RHL at x β 0
limβ¬(xβ1^+ ) f(x) = limβ¬(hβ0) f(1 + h)
= limβ¬(hβ0) 2(1+β)β1
= limβ¬(hβ0) 2+2ββ1
= 2 + 2(0) β 1
=2 β 1
= 1
& π(1) = 2(1)β1
π(π) = 1
Hence, L.H.L = R.H.L = π(1)
β΄ f is continuous at x = 1
Therefore, f is continuous at x = 0 as well as at x = 1.
So, the correct answer is (A)

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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.