Chapter 4 - Applications of the Derivative - 4.3 Graphing Functions - 4.3 Exercises - Page 268: 10

$$\eqalign{ & {\text{local minimum }}\left( { - 1, - 2} \right) \cr & {\text{local maximum }}\left( {1,2} \right) \cr & {\text{inflection point }}\left( {0,0} \right) \cr & y{\text{ - intercept }}\left( {0,0} \right) \cr & x{\text{ - intercepts }}\left( {0,0} \right){\text{ and }}\left( { \pm \sqrt 3 ,0} \right) \cr} $$

$$\eqalign{ & f\left( x \right) = 3x - {x^3} \cr & {\text{Differentiate}} \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {3x - {x^3}} \right] \cr & f'\left( x \right) = 3 - 3{x^2} \cr & {\text{Let }}f'\left( x \right) = 0{\text{ to find the critical points}} \cr & 3 - 3{x^2} = 0 \cr & 3{x^2} = 3 \cr & x = \pm 1 \cr & {\text{The critical points are }}x = - 1{\text{ and }}x = 1 \cr & \cr & {\text{Find the second derivative}} \cr & f''\left( x \right) = \frac{d}{{dx}}\left[ {3 - 3{x^2}} \right] \cr & f''\left( x \right) = - 6x \cr & \cr & {\text{Evaluate }}f''\left( x \right){\text{ at the critical points }}x = - 1{\text{ and }}x = 1 \cr & f''\left( { - 1} \right) = - 6\left( { - 1} \right) = 6 > 0,{\text{ then}} \cr & {\text{There is a local minimum at }}x = - 1 \cr & f\left( { - 1} \right) = 3\left( { - 1} \right) - {\left( { - 1} \right)^3} = - 2 \cr & {\text{local minimum }}\left( { - 1, - 2} \right) \cr & f''\left( 1 \right) = - 6\left( 1 \right) = - 6 < 0,{\text{ then}} \cr & {\text{There is a local maximum at }}x = 1 \cr & f\left( 1 \right) = 3\left( 1 \right) - {\left( 1 \right)^3} = 2 \cr & {\text{local maximum }}\left( {1,2} \right) \cr & \cr & {\text{Set }}f''\left( x \right) = 0{\text{ to locate the inflection points}} \cr & f''\left( x \right) = - 6x \cr & - 6x = 0 \cr & x = 0 \cr & f\left( 0 \right) = 3\left( 0 \right) - {\left( 0 \right)^3} = 0 \cr & {\text{The inflection point is }}\left( {0,0} \right) \cr & \cr & {\text{Find the }}y{\text{ intercept}}{\text{, let }}x = 0 \cr & f\left( 0 \right) = 3\left( 0 \right) - {\left( 0 \right)^3} \cr & f\left( 0 \right) = 0 \cr & y{\text{ - intercept }}\left( {0,0} \right) \cr & {\text{Find the }}x{\text{ intercept}}{\text{, let }}f\left( x \right) = 0 \cr & 0 = 3x - {x^3} \cr & x\left( {3 - {x^2}} \right) = 0 \cr & x = 0,{\text{ }}x = - \sqrt 3 ,{\text{ }}x = \sqrt 3 \cr & x{\text{ - intercepts }}\left( {0,0} \right){\text{ and }}\left( { \pm \sqrt 3 ,0} \right) \cr & \cr & {\text{Summary:}} \cr & {\text{local minimum }}\left( { - 1, - 2} \right) \cr & {\text{local maximum }}\left( {1,2} \right) \cr & {\text{inflection point }}\left( {0,0} \right) \cr & y{\text{ - intercept }}\left( {0,0} \right) \cr & x{\text{ - intercepts }}\left( {0,0} \right){\text{ and }}\left( { \pm \sqrt 3 ,0} \right) \cr & \cr & {\text{Graph}} \cr} $$