Chapter 4 - Applications of the Derivative - 4.1 Maxima and Minima - 4.1 Exercises - Page 243: 50

$$\eqalign{ & \left( {\text{a}} \right)x = - 3{\text{ and }}x = \frac{1}{2} \cr & \left( {\text{b}} \right){\text{absolute maximum of 11 at }}x = \pm 1 \cr & {\text{absolute minimum of }} - 16{\text{ at }}x = \pm 2 \cr & \left( {\text{c}} \right){\text{ graph}} \cr} $$

$$\eqalign{ & f\left( x \right) = 2{x^6} - 15{x^4} + 24{x^2}{\text{ on }}\left[ { - 2,2} \right] \cr & \cr & {\text{Differentiate}} \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {2{x^6} - 15{x^4} + 24{x^2}} \right] \cr & f'\left( x \right) = 12{x^5} - 60{x^3} + 48x \cr & \cr & \left( a \right){\text{ Set the derivative equals to 0}} \cr & f'\left( x \right) = 0 \cr & 12{x^5} - 60{x^3} + 48x = 0 \cr & {\text{Factoring}} \cr & 12x\left( {{x^4} - 5{x^2} + 4} \right) = 0 \cr & 12x\left( {{x^2} - 4} \right)\left( {{x^2} - 1} \right) = 0 \cr & {\text{The critical points are:}} \cr & x = 0,{\text{ }}x = \pm 1,{\text{ }}x = \pm 2 \cr & \cr & \left( {\text{b}} \right){\text{Evaluate }}f\left( x \right){\text{ at the critical points and the endpoints }} \cr & f\left( { - 2} \right) = 2{\left( { - 2} \right)^6} - 15{\left( { - 2} \right)^4} + 24{\left( { - 2} \right)^2} = - 16,{\text{ }}\left( {{\text{Smallest}}} \right) \cr & f\left( { - 1} \right) = 2{\left( { - 1} \right)^6} - 15{\left( { - 1} \right)^4} + 24{\left( { - 1} \right)^2} = 11,{\text{ }}\left( {{\text{Largest}}} \right) \cr & f\left( 0 \right) = 2{\left( 0 \right)^6} - 15{\left( 0 \right)^4} + 24{\left( 0 \right)^2} = 0 \cr & f\left( 1 \right) = 2{\left( 1 \right)^6} - 15{\left( 1 \right)^4} + 24{\left( 1 \right)^2} = 11,{\text{ }}\left( {{\text{Largest}}} \right) \cr & f\left( 2 \right) = 2{\left( 2 \right)^6} - 15{\left( 2 \right)^4} + 24{\left( 2 \right)^2} = - 16,{\text{ }}\left( {{\text{Smallest}}} \right) \cr & {\text{Therefore}}{\text{,}} \cr & {\text{*The absolute maximum of }}f\left( x \right){\text{ is 11 at }}x = - 1{\text{ and }}x = 1 \cr & {\text{*The absolute minimum of }}f\left( x \right){\text{ is }} - 16{\text{ at }}x = - 2{\text{ and }}x = 2 \cr & \cr & \left( {\text{c}} \right){\text{Graph}} \cr} $$