1. ## [urgent] curve sketching..

Hi, I'm stuck on my calculus problem which is due tonight.

Here's the full question,

Q. For the following functions, do a full analysis of the function itself, the first derivative, and the second derivative.

- domain, intercepts, and vertical and horizontal asymptotes.
Find and classify all critical points
Find all intervals where the function is increasing or decreasing.
Find all intervals of concavity, and inflection points.
Finally, sketch the graph of the function, using the information you obtained.

- Is this mean by I have to do the analysis of three function? since it says, i have to do the analysis of the function itself, the first derivative and second.

i) f(x) = x(x^2+4)^2

I got the first and second derivatives,
f'(x) = (x^2+4)(5x^2+4)
f''(x) = 4x(5x^2+12)

Since I tried by looking at the text book and do the analysis.. I dunno where to start and what to do.. Also, for getting critical points, let say where f(x) = 0, I can't find the roots since i get no real number..

ii) f(x) = x^(5/3) - 5x^(2/3)
i also have the derivatives for these too.
f'(x) = 5/3*x(2/3) - 10/3*x^(-1/3)
f''(x) = 10/9*x^(-1/3) + 10/9*x(-4/3)

for this one ii), I have to care for cusps or vertical tangents..

iii) f(x) = 2cosx+sin2x
also got the derivatives.
f'(x) = -2sin(x)+2cos(2x)
f''(x) = -2cos(x) - 4sin(2x) ==> is this right?

and for this one, I have to care about whether the function is periodic or not..

Please math people.. help me.. I'm a newb

2. i)
First of all, this function is a odd function, so it increase from negative infinity to positive infinite

inflection point @ X=0 (0,0) (Concave down ---> 0 ---> Concave up)

3. ii)
Critical number X=0 X=2
Decrease (negative infinity, 0), (0,2)
Increase (2, positive infinity)

inflection points X=0 X=-1
Concave down (negative infinity, -1)
Concave up (-1,0) (0, positive infinity)

4. periodic???