Question

We have `f:RR->RR,f(x)=|x|root(3)(1-x^2)`.How to find maximum domain of differentiability?

Answer 1

See below.

Explanation:

`f(x)=|x|root(3)(1-x^2) = { (xroot(3)(1-x^2),x >= 0),(-xroot(3)(1-x^2),x < 0) :}`

`d/dx(xroot(3)(1-x^2)) = root(3)(1-x^2) + x(1/3(1-x^2)^(-2/3)(-2x))`

`= root(3)(1-x^2) - (2x^2)/(3(1-x^2)^(2/3))`

`= (3(1-x^2)-2x^2)/(3(1-x^2)^(2/3))`

`= (3-5x^2)/(3(1-x^2)^(2/3))`. `" "` For `x != +-1`

So,

`f'(x)= { ((3-5x^2)/(3(1-x^2)^(2/3)),x > 0,x != 1),(-(3-5x^2)/(3(1-x^2)^(2/3)),x < 0,x != -1) :}`
And `f` is not differentiable at `+-1`

Checking the "joint" of the "hinge" of the two parts, we see that the right derivative at `0` is `1` and the left derivative at `0` is `-1`, so there is no derivative at `0`.

Finally, here is the graph of the function: graph{y=(1-x^2)^(1/3)absx [-2.433, 2.436, -1.215, 1.217]}