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Question

Let f be a twice differentiable function defined on R such that f(0)=1, f(0)=2 and f(x)0 for all xR. If f(x)f(x)f(x)f′′(x)=0, for all xR, then the value of f(1) lies in the interval

A
(9,12)
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B
(6,9)
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C
(3,6)
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D
(0,3)
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Solution

The correct option is B (6,9)
Given f(x)f′′(x)(f(x))2=0
Let h(x)=f(x)f(x)
Then h(x)=0h(x)=k
f(x)f(x)=k
f(x)=kf(x)
f(0)=kf(0)
k=12

Now, f(x)=12f(x)
2 dx=f(x)f(x)dx
2x=ln|f(x)|+C
As f(0)=1C=0
2x=ln|f(x)|
f(x)=±e2x
As f(0)=1f(x)=e2x
f(1)=e27.38

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