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Question


Let f:[12,2]R and g:[12,2]R be functions defined by
f(x)=[x23] and g(x)=|x|f(x)+|4x7|f(x),where [y] denotes the greatest integer less than or equal to y for yϵR. Then


A
f is discontinuous exactly at three points in [12,2]
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B
f is discontinuous exactly at four points in [12,2]
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C
g is not differentiable exactly at four points in (12,2)
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D
g is not differentiable exactly at five points in (12,2)
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Solution

The correct options are
B f is discontinuous exactly at four points in [12,2]
C g is not differentiable exactly at four points in (12,2)
f and g: [12,2]R
f(x)=[x23] and g(x)=|x|f(x)+|4x7|f(x)
[x23] is discontinuous at all integral points in
[12,2] which happens at x=1,x=2,x=3,x=2
F is discontinuous at 4 points.
g(x)=(|x|+|4x7|)f(x)
F is not differentiable at x=1,2,3
And |x|+|4x7| is not differentiable at x=0
x=74
f(x)=0 in [3,2]
So at 74ϵ[3,2] the f(x) is differentiable
Hence g(x) is not differentiable at 4 points

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