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Question

Consider f(x) = [1 x], 1 x2 and g(x) = f(x) + b sin π2x,1x2 then which of the following is correct ?

A
Rolles theorem is applicable to both f, g and b = 32
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B
LMVT is not applicable to f and Rolles theorem is applicable to g with b = 12
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C
LMVT is not applicable to f and Rolles theorem is applicable to g with b=1
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D
Rolles theorem is not applicable to both f, g for any real b.
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Solution

The correct option is D LMVT is not applicable to f and Rolles theorem is applicable to g with b=1
f(x)=[1x],1x2
g(x)=f(x)+bsinπ2x,1x2
LMVT is not applicable to f(x), as f(x) is a step greatest integer function and is not continuous.
g(x)=f(x)+bsinπ2x1x2
f(x) will given an integer, thus let it be a constant k
g(x)=k+bsinπ2x
and for b=1
g(x)=k+sinπ2x
sin(π2x) will be a continuous and differentiable function in 1x2 with period of sin(π2x)=2π
And thus g(x) follow Relle's theorem with b=1

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