Limits Continuity And Differentiability Question 46
Question: Which of the following function is continuous at for all value of x?
(i) $ f( x ) $ =sgn $ (x^{3}-x) $
(ii) $ f( x ) $ =sgn $ (2\cos x-1) $
(iii) $ f( x ) $ =sgn $ (x^{2}-2x+3) $
Options:
A) Only (i)
B) Only (iii)
C) Both (ii) and (iii)
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
(i) $ f( x ) $ =sgn $ (x^{3}-x) $
Here $ x^{3}-x=0\Rightarrow x=0,-1,1 $
Hence, $ f(x) $ is discontinuous at $ x=0,-1,1 $ . (ii)
If $ f( x ) $ =sgn $ (2\cos x-1) $
Here, $ 2\cos x-1=0 $
$ \Rightarrow \cos x=1/2\Rightarrow x=2n\pi +(\pi /3) $
$ n\in Z, $ where $ f(x) $ is discontinuous. (iii) $ f( x ) $ =sgn $ (x^{2}-2x+3) $
Here, $ x^{2}-2x+3>0 $ for all x.
Thus, $ f(x)=1 $ for all x
Hence continuous for all x.
BETA
