curve sketching: f(x) = (x^1/3)(x + 3)^2/3

[christie-kay]

Sketch the curve of: f(x) = (x^1/3)(x + 3)^2/3

I'm really confused on this problem, since it is not working out as well as the others. My teacher wants us to first find the domain, the intercepts, the symmetry (if any), the aymptotes, the max/min values, and the concavity. Here is what I have so far;

Domain:
x can equal any real number

Intercepts:
y-intercept:
Let x=0
y=(0^1/3)(0+3)^2/3
y=0

x-intercept:
Let y=0
0=(x^1/3)(x+3)^2/3
**I'm not sure what to do with the exponents?

Symmetry
f(-x)=(-x^1/3)(-x+3)^2/3
**Again, I'm not sure what to do. Am I suppose to factor or distribute?

Aysmptotes:
none.

Max/Min points:
f(x) = (x^1/3)(x + 3)^2/3
f'(x) = (1/3x^-2/3)(x + 3)^2/3 + (x^1/3)(2/3(x + 3)^-1/3)
f'(x) = 1/3(x + 3) + 2/3x
**Did Idifferentiate that right? It doesn't look right.

Thanks for any help; I really appreciate it! I apologize for the little work I did on this problem, but I'm really confused. I hate FRACTIONS!!

 

[stapel]

Multi-post.

 

[christie-kay]

thanks, but what does "cbrt"?

And how did that person get, [ x(x + 3)^2 ]?

 

[stapel]

thanks, but what does "cbrt"?

The one-third power is the same as the cube root. So "x^1/3" is the same as "cbrt[x]".

And how did that person get, [ x(x + 3)^2 ]?

By manipulating the derivative. (Tutors often don't do the entire assignment for students, instead providing set-up, hints, helps, and suggestions. Partial answers, such as what you should get part of the way through the exercise, enable the student to check his work as he goes along.)

Please reply to this or the other thread showing all of the progress you have made. Thank you.

Eliz.

 

[christie-kay]

I finally got the first derivative and here is my table from the points that i got.
x+1/x^2/3(x+3)^1/3

Therefore: f(-3)=0 (local min) and f(-1)= (local max)

As for the second derivative..
f"(x)=x^2/3(x+3)^1/3 - (x+1)(2/3x^-1/3)(1/3(x+3)^-2/3)/(x^2/3)(x+3)^1/3)^2



Im not sure how should factor out the exponents, i went on different websites but none seemed to help...