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Question

The function f(x) is given by f(x)=x2 sin (1x) ,       if x00 ,                    if x=0
Which of the following statement(s) is/are true for f(x)?
  1. f(x) is continuous at x = 0
  2. f(x) is differentiable at x = 0
  3. f(0)=0
  4. f(x) is not differentiable at x = 0


Solution

The correct options are
A f(x) is continuous at x = 0
B f(x) is differentiable at x = 0
C f(0)=0
Given  that
 f(x)=x2 sin (1x) ,       if x00 ,                    if x=0limx0 x2=0 and 1sin (1x)1x2x2sin (1x)x2limx0x2limx0 x2sin (1x)limx0 x2By Sandwitch Theorem, limx0 x2 sin (1x)=0f(x) is continuous at x=0.Now,(RHD at x=0)=limx0+ f(x)f(0)x0=limh0+ f(0+h)f(0)0+h0=limh0+ f(h)f(0)h=limh0+ (h)2 sin(1h)0h=limh0+ h sin (1h)=0 ×(an oscillating number between 1 and 1)(RHD at x=0)=0(LHD at x=0)=limx0 f(x)f(0)x0=limh0 f(0h)f(0)0h0=limh0 f(h)f(0)h=limh0 (h)2 sin(1h)0h=limh0 h sin (1h)=0 ×(an oscillating number between 1 and 1)(LHD at x=0)=0(LHD at x=0)=(RHD at x=0)So, f(x) is differentiable at x=0 and f'(0)=0

 

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