Chapter 4 - Applications of the Derivative - 4.3 Graphing Functions - 4.3 Exercises - Page 268: 40

$$\eqalign{ & y{\text{ - intercept }}\left( {0, - 2} \right) \cr & x{\text{ - intercepts: }}\left( {0.0099,0} \right),\left( {8.98,0} \right),\left( {24.00,0} \right) \cr & {\text{local maximum at }}\left( {4,398} \right) \cr & {\text{local minimum at }}\left( {18, - 974} \right) \cr & {\text{inflection points }}\left( {11, - 288} \right) \cr} $$

$$\eqalign{ & f\left( x \right) = {x^3} - 33{x^2} + 216x - 2 \cr & {\text{Domain: }}\left( { - \infty ,\infty } \right) \cr & \cr & {\text{Find the }}y{\text{ intercept}}{\text{, let }}x = 0 \cr & f\left( 0 \right) = {\left( 0 \right)^3} - 33{\left( 0 \right)^2} + 216\left( 0 \right) - 2 \cr & f\left( 0 \right) = - 2 \cr & y{\text{ - intercept }}\left( {0, - 2} \right) \cr & {\text{Find the }}x{\text{ intercept}}{\text{, let }}f\left( x \right) = 0 \cr & {x^3} - 33{x^2} + 216x - 2 = 0 \cr & {\text{Using the graphing utility we obtain:}} \cr & {x_1} \approx 0.0099,{\text{ }}{x_2} \approx 8.98,{\text{ }}{x_3} \approx 24.00 \cr & x{\text{ - intercepts: }}\left( {0.0099,0} \right),\left( {8.98,0} \right),\left( {24.00,0} \right) \cr & \cr & *{\text{Find the extrema}} \cr & {\text{Differentiate}} \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {{x^3} - 33{x^2} + 216x - 2} \right] \cr & f'\left( x \right) = 3{x^2} - 66x + 216 \cr & {\text{Let }}f'\left( x \right) = 0{\text{ to find critical points}} \cr & 3{x^2} - 66x + 216 = 0 \cr & {x^2} - 22x + 72 = 0 \cr & \left( {x - 4} \right)\left( {x - 18} \right) = 0 \cr & x = 4,{\text{ }}x = 18, \cr & \cr & *{\text{Find the second derivative}} \cr & f''\left( x \right) = \frac{d}{{dx}}\left[ {3{x^2} - 66x + 216} \right] \cr & f''\left( x \right) = 6x - 66 \cr & \cr & {\text{Evaluate }}f''\left( x \right){\text{ at the critical points }}x = 4,{\text{ }}x = 18 \cr & *f''\left( 4 \right) = 6\left( 4 \right) - 66 = - 42 < 0,{\text{ then}} \cr & {\text{There is a local maximum at }}\left( {4,f\left( 4 \right)} \right) \cr & f\left( 4 \right) = {\left( 4 \right)^3} - 33{\left( 4 \right)^2} + 216\left( 4 \right) - 2 = 398 \cr & \to {\text{local maximum at }}\left( {4,398} \right) \cr & *f''\left( {18} \right) = 6\left( {18} \right) - 66 = 42 > 0,{\text{ then}} \cr & {\text{There is a local minimum at }}\left( {18,f\left( {18} \right)} \right) \cr & f\left( {18} \right) = {\left( {18} \right)^3} - 33{\left( {18} \right)^2} + 216\left( {18} \right) - 2 = - 974 \cr & \to {\text{local minimum at }}\left( {18, - 974} \right) \cr & \cr & {\text{*Find the Inflection points}}{\text{, set }}f''\left( x \right) = 0 \cr & 6x - 66 \cr & x = 11 \cr & f\left( {11} \right) = {\left( {11} \right)^3} - 33{\left( {11} \right)^2} + 216\left( {11} \right) - 2 = - 288 \cr & {\text{The inflection points is: }} \cr & \left( {11, - 288} \right) \cr & \cr & {\text{Graph}} \cr} $$