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Question

Let f(x)={x3,x<0x2,x0 then

A
f(x) is continuous at x = 0
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B
f(x) is differentiable at x = 0
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C
f(x) is continuous on R
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D
f′′(x) exists for all x ϵ R
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Solution

The correct options are
A f(x) is continuous at x = 0
B f(x) is differentiable at x = 0
C f(x) is continuous on R
f(0+)=f(0)=f(0)
f(0+)=f(0)=f(0)=0
So, f(x) is continuous at x=0
f(x)={3x2,x<02x,x0
f(0+)=f(0)=0,f(0)=0
So, f(x) is differentiable and continuous at x=0 and f(x) is continuous but non differentiable at x=0. Hence, f′′(x) will not be defined at x=0
f′′(x)={6x,x<02,x>0
f"(0+)=2f"(0)=0
So, f′′(x) is not defined at x=0

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