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Question

Let f:RR be a function defined by
f(x)=sin(x2)xif x0 0if x=0
Then, at x=0, f is

A
not continuous
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B
continuous but not differentiable
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C
differentiable and the derivative is not continuous
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D
differentiable and the derivative is continuous
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Solution

The correct option is D differentiable and the derivative is continuous
For continuity,
limx0f(x)=limx0sin(x2)x=limx0sin(x2)x2×x=0
So the function is continuous at x=0
Checking the differentiability at x=0
Right hand derivative,
R.H.D =limh0f(0+h)f(0)h=limh0sin(h2)h0h=limh0sin(h2)h2=1
Left hand derivative,
L.H.D =limh0f(0h)f(0)h=limh0sin((h)2)h0h=limh0sin(h2)h2=1
So, L.H.D=R.H.D then function is differentiable at x=0 and f(0)=1
When x0
f(x)=d(sin(x2)x)dxf(x)=2x2cos(x2)sin(x2)x2f(x)=2cos(x2)sin(x2)x2f(x)=2cos(x2)sin(x2)x2if x01if x=0
Checking the continuity of f(x),
limx0f(x)=limx02cos(x2)sin(x2)x2=21=1
So the derivative is also continuous.

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