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Question

If f(x)={x2(sgn[x])+{x},0x<2sinx+|x3|,2x<4,
where {x} and [x] represent the fractional part function and greatest integer function respectively. Then

A
f(x) is differentiable at x=1.
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B
f(x) is continuous but non-differentiable at x=1.
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C
f(x) is non-differentiable at x=2.
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D
f(x) is continuous at x=2.
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Solution

The correct option is C f(x) is non-differentiable at x=2.
f(x)=⎪ ⎪ ⎪⎪ ⎪ ⎪x2(0)+{x},0x<1x2(1)+{x},1x<2sinxx+3,2x<3sinx+x3,3x<4

f(x)=⎪ ⎪⎪ ⎪1,0<x<12x+1,1<x<2cosx1,2<x<3cosx+1,3<x<4

Clearly, from the above defining of f(x) and f(x), we can conclude that f(x) is continuous at x=1 since f(1)=f(1+)=f(1)=1. But f(x) is non differentiable at x=1 since f(1+)=3 but f(1)=1.
Also, at x=2, f(2+)=sin2+1 but f(2)=5
So, f(x) is discontinuous at x=2 and hence f(x) is non-differentiable at x=2.

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