Question

Let f(x) = x |x|, for all x ∈ R check its differentiability at x = 0.

Answer 1

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

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

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

= 0

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

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

= 0

R.f'(0) = L.f'(0)

Therefore, f(x) is differentiable at x = 0.