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Question

If f(x)=x3sgn(x), then

A
f is differentiable at x=0.
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B
f is continous but not differentiable at x=0.
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C
f(0)=1
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D
None of these
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Solution

The correct option is A f is differentiable at x=0.
f(x)=x3sgn(x)f(x)=x3,x>00,x=0x3,x<0
Only point at which f(x) may be discontinuous is x=0
limx0f(x)=limx0+f(x)=f(0)=0
So, f(x) is continuous at x=0

Now,
f(x)={3x2,x>03x2,x<0f(0)=f(0+)=0
So, f(x) is differentiable at x=0

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