Q:

Analyze and sketch the graph of $f(x) = x^4 - 12x^3 + 48x^2 - 64x$

I'm currently doing AP Calculus AB homework, and need some help with this problem. It sucks to forget. I know how to get the first and second derivative. I know that critical numbers can be found using the first derivative, and that points of inflection and concavity intervals can be found by using the second derivative, but I honestly forgot how to do it. 
Analyze and sketch the graph of $f(x) = x^4 - 12x^3 + 48x^2 - 64x$
Edit: Solved!

First Derivative: $4x^3 - 36x^2 + 96x - 64$
Second Derivative: $12x^2 - 72x + 96$
X-Intercepts: $(0,0)$ $(4,0)$
Y-Intercepts: $(0,0)$
Critical Numbers: $X = 1$, $X = 4$
Points of Inflection: $X = 2$, $X = 4$
Increasing Intervals: $(1, Infinity)$
Decreasing Intervals: $(-Infinity, 1)$
Concavity Intervals: $(-Infinity, 2)$ $(2,4)$ $(4, Infinity)$
Relative Minimums: $X=1$
Relative Maximums: $X=2$

A:

This should bring you forward: 

X- Intercepts: f(x) =0
Y- Intercepts: f(0)
critical numbers: f'(x) = 0 
Points of inflections: f''(x) = 0 and f'''(x) $\neq$ 0
Increasing intervals: f'(x) > 0
Decreasing intervals: f'(x) < 0
Concavity upward: f''(x) > 0
Concavity downward f''(x) < 0
Relative Minimums: f'(x) = 0 (see above: critical numbers) and f''(x)>0
Relative Maximums: f'(x) = 0 (see above: critical numbers) and f''(x)<0

However, you should - at least once - really understand those criteria (e.g. graphically) rather than just apply them. 

