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Question

# Given f(x)=cos−1(sgn(2|x|3x−|x|)) where sgn(.) denotes the signum function [⋅] denotes the greatest integer function. Discuss the continuity and differentiablity at x=±1

A
f is continuous and derivable at x=1 but f is continuous but not derivable at x=1
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B
f is continuous but not derivable at x=1 but f is neither continuous nor derivable at x=1
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C
f is continuous and derivable at x=1,1
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D
f is continuous and derivable at x=1 but f is neither continuous nor derivable at x=1
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Solution

## The correct option is C f is continuous and derivable at x=−1 but f is neither continuous nor derivable at x=1f(1−)=limh→0cos−1(sgn(2[1−h]3(1−h)−[1−h]))=π2f(1+)=limh→0cos−1(sgn(2[1+h]3(1+h)−[1+h]))=limh→0cos−1(sgn(22))=0Hence f(x) isnot continuous and not dirivable at x=1Now at x=−1f(−1−)=limh→0cos−1(sgn(2[−1−h]3(−1−h)−[−1−h]))=limh→0cos−1(sgn(−4−3+2))=cos−11=0Alsof(−1+)=limh→0cos−1(sgn(2[−1+h]3(−1+h)−[−1+h]))=limh→0cos−1(sgn(−2−3+1))=cos−11=0Hence f(x) is continuous and differentiable at x=−1

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