How do you find the critical numbers for root3((x^2-x))3(x2x) to determine the maximum and minimum?

1 Answer
Jun 8, 2017

The critical number is x=1/2x=12.

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)(x2x)13

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/dxddx == 1/3(x^2-x)^(-2/3)(2x-1)13(x2x)23(2x1)

Now we rewrite it:

d/dxddx == (2x-1)/(3root(3)((x^2-x)^2)2x133(x2x)2

Set it equal to zero and solve:

(2x-1)/(3root(3)((x^2-x)^2)2x133(x2x)2 =0=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]}