Let `f (x)=xabsx,AAx in R`. Then,

Options

A. f is derivable at x = 0
B. f is not derivable at x = 0
C. f is not continuous at x = 0
D. None of the above

Answer

A

Explanation

The given function f is defined by
`f(x) ={ {:(s^(2)",",if x ge 0),(- x^(2)",", if x lt 0):}`
Now, LDH = ` underset(x to 0^(-)) lim (f(x) - f(0))/(x-0) = underset(x to 0^(-)) lim (-x^(2) -0)/(x)`
`= underset(x to 0^(-))lim - x = 0 `
`RHD underset(x to 0^(+)) lim (f(x) - f(0))/(x-0) = underset(x to 0^(+)) lim (-x^(2) -0)/(x-0)`
`= underset(x to 0^(+))lim x = 0 `
Since , LHD = RHD
`therefore f ` is derivable at x = 0 .