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Question

Two functions f(x) and g(x) are defined as f(x)={[x]{x} ,x00 ,x=0 and g(x)={{x}[x] ,x00 ,x=0 (where [.] and {.} refer to the greatest integer function and the fractional part function respectively and nN). Then which of the following statements is/are TRUE?

A
n+10f(x)dxn0f(x)dx=n1lnn
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B
n+10g(x)dxn0g(x)dx=1n+1
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C
t0f(x)dx=t0g(x)dx has at least one solution in t(0,π2)
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D
e<limnn0g(x)dx<e2
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Solution

The correct option is B n+10g(x)dxn0g(x)dx=1n+1
n+10f(x)dxn0f(x)dx=n+1nf(x)dx
=n+1nnxndx=1nn[nxlnn]n+1n
=n1lnn

n+10g(x)dxn0g(x)dx=n+1nf(x)dx
=n+1n(xn)ndx=[(xn)n+1n+1]n+1n
=1n+1

t(0,1)t0f(x)dx=0
t[1,π2)t0f(x)dx<π21
t(0,1)t0g(x)dx=1
t[1,π2)t0g(x)dx>0,
Hence, t0f(x)dx=t0g(x)dx t(0,π2), has no solution


limnn0g(x)dx=limnni=01i+1
It is divergent series.


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