Question

# The function f(x) is given by f(x)=⎛⎝x2 sin (1x) ,       if x≠00 ,                    if x=0 Which of the following statement(s) is/are true for f(x)?f(x) is continuous at x = 0f(x) is differentiable at x = 0f′(0)=0f(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)=0Given  that  f(x)=⎛⎝x2 sin (1x) ,       if x≠00 ,                    if x=0limx→0 x2=0 and −1≤sin (1x)≤1−x2≤x2sin (1x)≤x2limx→0−x2≤limx→0 x2sin (1x)≤limx→0 x2By Sandwitch Theorem, limx→0 x2 sin (1x)=0∴f(x) is continuous at x=0.Now,(RHD at x=0)=limx→0+ f(x)−f(0)x−0=limh→0+ f(0+h)−f(0)0+h−0=limh→0+ f(h)−f(0)h=limh→0+ (h)2 sin(1h)−0h=limh→0+ h sin (1h)=0 ×(an oscillating number between −1 and 1)⇒(RHD at x=0)=0(LHD at x=0)=limx→0− f(x)−f(0)x−0=limh→0− f(0−h)−f(0)0−h−0=limh→0− f(−h)−f(0)−h=limh→0− (−h)2 sin(1−h)−0−h=limh→0− 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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