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Question

If f(x)=xpcos1x,x00,x=0, then at x=0,f(x) is

A
continuous if p>0
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B
differentiable if p>1
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C
continuous if p>1
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D
differentiable if p>0
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Solution

The correct options are
A continuous if p>0
B differentiable if p>1
Continuity at x=0
L.H.L.=limh0f(0h)=limh0(h)pcos1h=0 if p>0
R.H.L.=limh0f(0+h)=limh0hpcos1h=0 if p>0
and,f(0)=0.
f(x) is continuous at x=0 if p>0
Lf(0)=limh0f(0h)f(0)h=limh0(h)pcos1h0h
=limh0(h)p1cos1h=0 if p>0,
and, Rf(0)=limh0f(0+h)f(0)h=limh0hpcos1h0h
=limh0hp1cos1h=0 if p>0
f(x) is differentiable at x=0 if p>1.

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