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Question

Let f(x)=1xx3. Find all real values of x satisfying the inequality,1f(x)f3(x)>f(15x)

A
(2,0)(2,)
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B
(0,2)
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C
(,0)(2,)
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D
(0,2)(2,)
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Solution

The correct option is D (2,0)(2,)
1f(x)f(x)2>f(15x)
Or
f(f(x))>f(15x) ...(i)
Now
f(x)=13x2
Hence
f(x)<0 for all x. ...(ii)
Hence
f(f(x)>f(15x)
But f(x) is a decreasing function (from ii).
Hence
f(x)<15x
Or
1xx3<15x.
Or
4xx3<0
Or
x34x>0
Or
x(x24)>0
Or
x>0 and x24>0 or
x<0 and x24<0
Hence if x>0 x24>0 implies xϵ(2,)
And
x<0 and x24<0 implies (2,0)
Hence
xϵ(2,0)(2,).

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