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A function f:RtoR is defined as f(x)=x^(...

A function `f:RtoR` is defined as `f(x)=x^(2)" for "xge0,f(x)=-x" for "xlt0`.
Consider the following statements in respect of the above function :
1. The function is continuous at x=0.
2. The function is differentiable at x=0.
Which of the above statements is /are correct ?

A

1 only

B

2 only

C

Both 1 and 2

D

Neither 1 nor 2

Text Solution

Verified by Experts

The correct Answer is:
A
`f:RtoR,f(x)={{:(x^(2)",",xge0),(-x",",xlt0):}`
For continuity at x=0
`f(0-0)=underset(hto0)limf(0-h)`
`=underset(hto0)lim[(0-h)]=underset(hto0)limh=0`
`f(0+0)=underset(hto0)limf(0+h)=underset(hto0)lim(0+h)^(2)=0andf(0)=0`
f(x) is continuous at x=0
f(x) is continuous at x=0
For differentiability at x=0
`underset(hto0)lim(-(h)-0)/(-h)=underset(hto0)lim=(h)/(-h)=-1`
and `underset(hto0)lim(f(0+h)-f(0))/(h)=underset(hto0)limh=0`
f(x) is not differentiable at x-0
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