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Question

Let F:RR be a thrice differentiable function. Supose that F(1)=0,F(3)=4 and F(x)<0 for all x(1/2,3). Let f(x)=xF(x) for all xR.

The correct statement(s) is(are)

A
f(1)<0
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B
f(2)<0
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C
f(x)0 for any x(1,3)
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D
f(x)=0 for some x(1,3)
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Solution

The correct option is C f(x)0 for any x(1,3)
f(x)=xF(x)f(x)=xF(x)+F(x)f(1)=F(1)+F(1)=F(1)f(1)<0

f(2)=2F(2)
As its is given F(x)<0 xR
So F(x) is decreasing in nature.
F(2)<F(1)F(2)<0
f(2)<0

f(x)=xF(x)+F(x)
We know that F(x)0 and xF(x)<0 x(1,3)

So f(x)<0 x(1,3)

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