The function `f(x)=[x^(2)]+[-x]^(2),` where [.] is GIF is

Options

A. continuous and derivable at x=2
B. neither continuous nor derivable at x=2
C. continuous but not dervable at x=2
D. none of these

Answer

B

Explanation

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.