Question

How do you find the intervals of increasing and decreasing using the first derivative given `y=(x-1)^(1/3)`?

Answer 1

Differentiate.

Let `y = u^(1/3)` and `u = x- 1`.

`y' = 1/3u^(-2/3)` and `u' = 1`

`dy/dx = 1/(3u^(2/3))`

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

The derivative will have a critical value of `1`, since it renders the derivative undefined. Check to the left and right of this point.

`dy/dx = 1/(3(3 - 1)^(2/3)) = 1/(3(root(3)(4))) ->" positive: the function is " color(blue) "increasing"`

`dy/dx = 1/(3(-2 - 1)^(2/3)) = 1/(3(root(3)(9))) ->" positive: the function is "color(blue) "increasing"`

So, the function `y = (x- 1)^(1/3)` is uniformly increasing.

Hopefully this helps!