Chapter 3 - Applications of Differentiation - 3.6 Exercises - Page 212: 23

$$\eqalign{ & {\text{Domain: }}\left( { - \infty ,\infty } \right) \cr & x{\text{ - intercepts }}\left( {0,0} \right),\left( { \pm \root 4 \of 5 ,0} \right) \cr & y{\text{ - intercept }}\left( {0,0} \right) \cr & {\text{Relative maximum }}\left( { - 1,4} \right) \cr & \left( {1, - 4} \right){\text{ Relative minimum}} \cr & {\text{Inflection point }}\left( {0,0} \right) \cr & {\text{No vertical asymptotes}} \cr & {\text{No horizontal asymptotes}} \cr} $$

$$\eqalign{ & y = {x^5} - 5x \cr & {\text{The domain of the function is: }}\left( { - \infty ,\infty } \right) \cr & \cr & {\text{Find the }}y{\text{ intercept}}{\text{, let }}x = 0 \cr & y = {\left( 0 \right)^5} - 5\left( 0 \right) \cr & y = 0 \cr & y{\text{ - intercept }}\left( {0,0} \right) \cr & \cr & {\text{Find the }}x{\text{ intercepts}}{\text{, let }}y = 0 \cr & {x^5} - 5x = 0 \cr & x\left( {{x^4} - 5} \right) = 0 \cr & x = 0,{\text{ }}x = \pm \root 4 \of 5 \cr & x{\text{ - intercepts }}\left( {0,0} \right),\left( { \pm \root 4 \of 5 ,0} \right) \cr & \cr & {\text{*Differentiate to find the critical points}} \cr & y' = \frac{d}{{dx}}\left[ {{x^5} - 5x} \right] \cr & y' = 5{x^4} - 5 \cr & {\text{Let }}y' = 0{\text{ to find critical points}} \cr & 5{x^4} - 5 = 0 \cr & 5{x^4} = 5 \cr & x = - 1,{\text{ }}x = 1 \cr & \cr & *{\text{Find the second derivative}} \cr & y'' = \frac{d}{{dx}}\left[ {5{x^4} - 5} \right] \cr & y'' = 20{x^3} \cr & {\text{Evaluate }}y''\left( x \right){\text{ at }}x = - 1,{\text{ }}x = 1 \cr & y''\left( { - 1} \right) = 20{\left( { - 1} \right)^3} = - 20 < 0,{\text{ Relative maximum at }}x = - 1 \cr & y\left( { - 1} \right) = {\left( { - 1} \right)^5} - 5\left( { - 1} \right) = 4,{\text{ }}\left( { - 1,4} \right){\text{ Relative maximum}} \cr & y''\left( 1 \right) = 20{\left( 1 \right)^3} = 20 < 0,{\text{ Relative minimum at }}x = - 1 \cr & y\left( 1 \right) = {\left( 1 \right)^5} - 5\left( 1 \right) = - 4,{\text{ }}\left( {1, - 4} \right){\text{ Relative minimum}} \cr & \cr & {\text{Let }}y''\left( x \right) = 0 \cr & 20{x^3} = 0 \cr & x = 0 \cr & {\text{Inflection points at }}x = 0 \cr & f\left( 0 \right) = 0 \cr & {\text{The inflection point is }}\left( {0,0} \right) \cr & \cr & {\text{*Calculate the asymptotes}} \cr & {\text{No vertical asymptotes}}{\text{, the denominator is 1}}{\text{.}} \cr & \mathop {\lim }\limits_{x \to \infty } \left( {{x^5} - 5x} \right) = \infty \cr & \mathop {\lim }\limits_{x \to - \infty } \left( {{x^5} - 5x} \right) = - \infty \cr & {\text{No horizontal asymptotes}} \cr & \cr & {\text{Graph}} \cr} $$