1.

The function `f(x)=[x^(2)]+[-x]^(2)`, where [.] denotes the greatest integer function, isA. continuous and derivable at x=2B. neither continuous nor derivable at x=2C. continuous but not dervable at x=2D. none of these

Answer» Correct Answer - B
We have
`underset(x to 2^(-))lim f(x)=underset(h to 0)lim f(2-h)=underset(h to 0)lim [(2-h)^(2)]+[-2+h]^(2)`
`Rightarrow underset(x to 2^(-))lim f(x)=3+(-2)^(2)=7`
`underset(x to 2^(+))lim f(x)=underset(h to 0)lim f(2+h)=underset(h to 0)lim [(2+h)^(2)]+[-2-h]^(2)`
`Rightarrow underset(x to 2^(-))lim f(x) ne underset(x to 2^(+))lim f(x)`
So, f(x) is discontinuous at x=2
Consequently, it is non -differentiable at x=2.


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