How to Find Extrema, Roots, Inflection Points, and Concavity Using Derivatives?

[crybllrd]

Homework Statement

$y=3(x-1)^{\frac{1}{3}}-(x-1)^{2}$

I need to find all extrema, roots, inflection points, and concavity using first and second derivative tests.
I usually do not have a problem with these, but I need to find some extrema that I know exist.

The Attempt at a Solution

$y'=2(x-1)^{\frac{-1}{3}}-2(x-1)$

simplified to:

$y'=\frac{2+(-2x+2)(x-1)^{\frac{1}{3}}}{(x-1)^{\frac{1}{3}}}$

From looking at a graph I can see that there are extrema at x=0, and x=2. There is also a sharp turn at (1,0). However, i am not seeing these extrema in my first derivative. Did I not simplify enough or make an algebraic error?

 

[Mark44]

Homework Statement

$y=3(x-1)^{\frac{1}{3}}-(x-1)^{2}$

I need to find all extrema, roots, inflection points, and concavity using first and second derivative tests.
I usually do not have a problem with these, but I need to find some extrema that I know exist.

The Attempt at a Solution

$y'=2(x-1)^{\frac{-1}{3}}-2(x-1)$

There's a mistake in the above.
d/dx(3(x - 1)^(1/3)) = (x - 1)^(-2/3)

simplified to:

$y'=\frac{2+(-2x+2)(x-1)^{\frac{1}{3}}}{(x-1)^{\frac{1}{3}}}$

From looking at a graph I can see that there are extrema at x=0, and x=2. There is also a sharp turn at (1,0). However, i am not seeing these extrema in my first derivative. Did I not simplify enough or make an algebraic error?

 

[crybllrd]

Thanks, I'm not sure how I missed that. I can take it from here.
Thanks again