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Question

Let f:(0,)(0,) be a differentiable function such that f(1)=e and limtxt2f2(x)x2f2(t)tx=0. If f(x)=1, then x is equal to

A
e
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B
2e
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C
1e
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D
12e
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Solution

The correct option is C 1e
limtxt2f2(x)x2f2(t)tx=0
Usingf L'Hospital rule
limtx2tf2(x)x2(2f(t))f(t)1=0
2xf2(x)2x2f(x)f(x)=0
2xf(x)(f(x)xf(x))=0
f(x)f(x)=1x (x0,f(x)0)
lnf(x)=ln(x)+lnC
f(x)=Cx
If x=1C=e
f(x)=ex
If f(x)=1x=1e

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