Question

Determine whether the following function is differentiable at the indicated values.

f(x) = x |x| at x = 0

Answer 1

`f(x) = {{:(x(- x),  "if" x < 0),(x(x),  "if"  x > 0):}`

`f(x) = {{:(- x^2,  "if" x < 0),(x^2,  "if"  x > 0):}`

To find the left limit of `f(x)` at x = 0

When x → 0

`f(x) = - x^2`

`f"'"(0^-) =  lim_(x -> 0^-) (f(x) - f(0))/(x - 0)`

= `lim_(x -> 0^-) (-x^2 - 0^2)/(x - 0)`

= `lim_(x -> 0^-) (-x^2)/x`

= `lim_(x -> 0^-) (- x)` = 0  .........(1)

To find the right limit of `f(x)` at x = 0

When x → 0⁺

`f(x) = x^2`

`f"'"(0^+) =  lim_(x -> 0^+) (f(x) - f(0))/(x - 0)`

= `lim_(x -> 0^+) (x^2 - 0^2)/x`

= `lim_(x -> 0^+) (x^2)/x`

= `lim_(x -> 0^+) x` = 0  .........(2)

From eqation (1) and (2) we  get

`f"'"(0-)  =  f"'"(0^+)`

∴ f(x) is differentiable at x = 0.