Question

How do you find the critical numbers for `root3((x^2-x))` to determine the maximum and minimum?

Answer 1

The critical number is `x=1/2`.

Explanation:

First we take the derivative using the chain rule, to make things easier for us we rewrite the problem using powers.

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

Now we apply the chain rule we take the derivative of the outside and multiple it by the derivative of the inside. It's important that you know the power rule.

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

Now we rewrite it:

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

Set it equal to zero and solve:

`(2x-1)/(3root(3)((x^2-x)^2)` `=0`

`cancelcolor(green)(3root(3)((x^2-x)^2))/1**(2x-1)/cancelcolor(green)(3root(3)((x^2-x)^2)` `=0` `(3root(3)((x^2-x)^2))`

We are left with:

`2x-1=0`

Solve:

`x=1/2`

By looking at the graph you can see that `x=1/2` is a critical number.
graph{(x^2-x)^(1/3) [-1.912, 4.248, -1.023, 2.057]}