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 [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{(−∞, 0) ∪ (2, +∞)} [1.5] 1h.) f decreasing on the intervals \underline{(0, 2)} [1.5] 1i.) f is concave up on the intervals \underline{(1, +∞)} [1.5] 1j.) f is concave down on the intervals\underline{(−∞, 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} [4.5] 1m.) Graph f 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).

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).