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Question

Let f be defined on R by f(x)={x4sin(1/x);x00;x=0, then

A
f(0) doesn't exist
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B
f′′(0) doesn't exist
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C
f′′(x) is continuous at x=0
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D
f′′(0) does not exist but f′′(x) is not continuous at x=0
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Solution

The correct option is D f′′(0) does not exist but f′′(x) is not continuous at x=0
f(x)={x4sin1xx00x=0
For x0
f(x)=3x2sin1x+x4cos(1x)(1x2)
=3x2sin1xx2cos1x
Now
f(x)f(0)x0=x4sin1xx=x3sin1xf(0)=0
f′′(x)=9x2sin1x+3x2cos1x(1x2)
+2xcos1x3cos1x+2xcos1x+sin1x
f′′(0)
does not exits
And hence not continuous at x=0

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