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Question

On the real line R, we define two functions f and g as follows:
f(x)=min{x[x],1x+[x]},
g(x)=max{x[x],1x+[x]},
where [x] denotes the largest integer not exceeding x. The positive integer n for which n0(g(x)f(x)dx=100 is

A
100
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B
198
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C
200
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D
202
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Solution

The correct option is C 200
I will denote f(x) as f and g(x) as g.
frac(x)= fractional part of x.
frac(x)=x[x]
f(x)=min{x[x],1x+[x]}
f(x)=min{frac(x),1frac(x)}
g(x)=max{x[x],1x+[x]}
g(x)=max{frac(x),1frac(x)}
For 0frac(x)0.5,f=frac(x);g=1frac(x)
gf=12frac(x)
Similarly,For 0.5frac(x)1,g=frac(x);f=1frac(x)
gf=2frac(x)1
Given that,
n0g(x)f(x)dx=100
Since, the above function is periodic, i.e. P(k)=P(k+1) where P(K)=k+1kg(x)f(x)dx
10(g(x)f(x)dx=21g(x)f(x)dx=32(g(x)f(x)dx=nn1g(x)f(x)dx

n0(g(x)f(x)dx=n10g(x)f(x)dx n10g(x)f(x)dx=n(0.50g(x)f(x)dx+10.5g(x)f(x)dx)=100
n(0.5012frac(x)dx+10.52frac(x)1dx)=100
In the range from 0 to 1, frac(x)=x
n(0.5012xdx+10.52x1dx)=100
n(10.52xdx0.502xdx)=100
n=200


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