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Question

A function f(x) satisfying 10f(tx)dt=nf(x), where x>0 is

A
f(x)=c.x1nn
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B
f(x)=c.xnn1
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C
f(x)=c.x1n
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D
f(x)=c.x(1n)
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Solution

The correct option is A f(x)=c.x1nn
10f(tx)dt=10f(tx)d(tx)x=1xx0f(k)dk=nf(x)
x0f(k)dk=nxf(x)

On differentiating the above equation,
f(x)=nf(x)+nxf(x)
Let y=f(x)
yny=nxdydx
(1n)dxnx=dyy

On Integrating Both sides,
lny=1nnlnx+lncy=f(x)=cx1nn

So, Option A is correct answer.

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