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Question

Let f:0,0, be a differentiable function such that f1=e and limtxt2f2x-x2f2tt-x=0.if fx=1then 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


Explanation for the correct options:

Find the value of x.

Step 1: Given data

limtxt2f2x-x2f2tt-x=0

f:0,0,

f1=e

fx=1

Consider the given equation as

limtxt2f2x-x2f2tt-x=0x2f2x-x2f2xx-x=00

Step 2: Using the L 'Hospitality rule

limtx2tf2x-x22ftf't1=02xf2x-x22fxf'x=02xf2x=x22fxf'xfx=xf'xf'xfx=1x

Step 3: Integrate the above Equation

f'xfxdx=1xdxlogfx=logx+logeclogfx=logxecfx=xecf1=1ec[puttingx=1;given,f1=e]f1=ec=1

Then,

fx=xefx=1[givenf(x)=1]xe=1x=1e

Hence, the correct answer is option C.


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