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Find the following for $f(x) = x^3 - 3x^2 + 33$ (if 
they 
exist; if they don't exist, state so).  Use this information to graph $f$.

Note $f'(x) = {3x^2 - 6x}$, $f''(x) = {6x - 6}$, $f(-2) = 13$, $f(-3) = -21$ 

%%Is $f$ even, odd, periodic?  What is the domain and range of $f$?
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[1.5] 1a.) critical numbers: $\underline{0, 2}$
		

[1.5] 1b.) local maximum(s) occur at $x = \underline{0}$

[1.5] 1c.) local minimum(s) occur at $x = \underline{2}$

[1.5] 1d.)  The global maximum of $f$ on the interval [0, 5] is $\underline{83}$ and occurs at $x = \underline{5}$

[1.5] 1e.)  The global minimum of $f$ on the interval [0, 5] is $\underline{29}$ and occurs at $x = \underline{2}$

[1.5] 1f.) Inflection point(s) occur at $x = \underline{1}$


[1.5] 1g.)  $f$ increasing on the intervals $\underline{(-\infty, 0) \cup (2, +\infty)}$

[1.5] 1h.)  $f$ decreasing on the intervals $\underline{(0, 2)}$

[1.5] 1i.)  $f$ is concave up on the intervals $\underline{(1, +\infty)}$

[1.5] 1j.)  $f$ is concave down on the intervals$\underline{(-\infty, 1)}$

[1.5] 1k.)  Equation(s) of vertical asymptote(s)$\underline{none}$

[4] 1l.)  Equation(s) of horizontal and/or 
slant asymptote(s)$\underline{none}$


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[4.5] 1m.)  Graph $f$
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$f(x) = x^3 - 3x^2 + 33$ 

$f'(x) = {3x^2 - 6x} = 3x(x - 2) = 0 $ or DNE,  critical points: $x = 0, 2$

Check increasing/decreasing between critical points (f'(x) = 0, DNE)
and 
singleton points not in domain (where function could change between increasing/decreasing).

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$f''(x) = {6x - 6} = 0 $ or DNE, so possible inflection point:  $x = 1$

Check concave up/down between possible inflection points (f''(x) = 0, DNE) and singleton points not in domain (where function could 
change between concave up/down).


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