Chapter 3 - Applications of Differentiation - 3.6 Exercises - Page 212: 18

$$\eqalign{ & {\text{Domain: }}\left( { - \infty ,\infty } \right) \cr & y{\text{ - intercept }}\left( {0, - 2} \right) \cr & x{\text{ - intercepts }}\left( { - 1,0} \right),\left( {{3^{3/4}} - 1,0} \right),\left( {1 - {3^{3/4}},0} \right) \cr & {\text{Relative minimum at the points }}\left( { - 2,2} \right){\text{ and }}\left( {0, - 2} \right) \cr & {\text{No inflection point}} \cr & {\text{No vertical asymptotes}} \cr & {\text{No horizontal asymptotes}} \cr} $$

$$\eqalign{ & y = {\left( {x + 1} \right)^2} - 3{\left( {x + 1} \right)^{2/3}} \cr & \cr & {\text{*Domain: }}\left( { - \infty ,\infty } \right) \cr & \cr & {\text{*Find the }}y{\text{ intercept, let }}x = 0 \cr & y = {\left( {0 + 1} \right)^2} - 3{\left( {0 + 1} \right)^{2/3}} \cr & y = - 2 \cr & y{\text{ - intercept }}\left( {0, - 2} \right) \cr & \cr & {\text{*Find the }}x{\text{ intercept, let }}y = 0 \cr & {\left( {x + 1} \right)^2} - 3{\left( {x + 1} \right)^{2/3}} = 0 \cr & {\left( {x + 1} \right)^{2/3}}\left[ {{{\left( {x + 1} \right)}^{4/3}} - 3} \right] = 0 \cr & {\left( {x + 1} \right)^{2/3}} = 0 \to x = - 1 \cr & {\left( {x + 1} \right)^{4/3}} - 3 = 0 \cr & {\left( {x + 1} \right)^{4/3}} = 3 \cr & x = \pm \left( {{3^{3/4}} - 1} \right) \to x = 1 - {3^{3/4}},{\text{ }}x = {3^{3/4}} - 1 \cr & x{\text{ - intercepts }}\left( { - 1,0} \right),\left( {{3^{3/4}} - 1,0} \right),\left( {1 - {3^{3/4}},0} \right) \cr & \cr & *{\text{Find the first and second derivatives}} \cr & {\text{Differentiate}} \cr & y' = \frac{d}{{dx}}\left[ {{{\left( {x + 1} \right)}^2} - 3{{\left( {x + 1} \right)}^{2/3}}} \right] \cr & y' = 2\left( {x + 1} \right) - 2{\left( {x + 1} \right)^{ - 1/3}} \cr & {\text{Let }}y' = 0{\text{ to find critical points}} \cr & 2\left( {x + 1} \right) - 2{\left( {x + 1} \right)^{ - 1/3}} = 0 \cr & 2{\left( {x + 1} \right)^{ - 1/3}}\left[ {{{\left( {x + 1} \right)}^{4/3}} - 1} \right] = 0 \cr & 2{\left( {x + 1} \right)^{ - 1/3}} = 0,{\text{ undefined at }}x = - 1 \cr & {\left( {x + 1} \right)^{4/3}} - 1 = 0 \cr & {\left( {x + 1} \right)^{4/3}} = 1 \cr & x + 1 = \pm 1 \cr & {x_1} = - 2,{\text{ }}{x_2} = 0 \cr & \cr & y'' = \frac{d}{{dx}}\left[ {2\left( {x + 1} \right) - 2{{\left( {x + 1} \right)}^{ - 1/3}}} \right] \cr & y'' = 2 + \frac{2}{3}{\left( {x + 1} \right)^{ - 4/3}} \cr & {\text{Evaluate }}y''\left( x \right){\text{ at the critical points}} \cr & y''\left( { - 2} \right) = 2 + \frac{2}{3}{\left( { - 2 + 1} \right)^{ - 4/3}} = \frac{8}{3} > 0,{\text{ Relative minimum}} \cr & y\left( { - 2} \right) = - 2 \to {\text{Relative minimum at the point }}\left( { - 2,2} \right) \cr & y''\left( 0 \right) = 2 + \frac{2}{3}{\left( {0 + 1} \right)^{ - 4/3}} = \frac{8}{3} > 0,{\text{ Relative minimum}} \cr & y\left( 0 \right) = - 2 \to {\text{Relative minimum at the point }}\left( {0,-2} \right) \cr & \cr & {\text{Let }}y''\left( x \right) = 0 \cr & 2 + \frac{2}{3}{\left( {x + 1} \right)^{ - 4/3}} = 0 \cr & \frac{2}{3}{\left( {x + 1} \right)^{ - 4/3}} = - 2 \cr & {\text{No real solution, then there is no inflection point}} \cr & \cr & {\text{*Calculate the asymptotes}} \cr & {\text{No vertical asymptotes, the denominator is 1}}{\text{.}} \cr & \mathop {\lim }\limits_{x \to \infty } \left( {{{\left( {x + 1} \right)}^2} - 3{{\left( {x + 1} \right)}^{2/3}}} \right) = \infty \cr & \mathop {\lim }\limits_{x \to - \infty } \left( {{{\left( {x + 1} \right)}^2} - 3{{\left( {x + 1} \right)}^{2/3}}} \right) = \infty \cr & {\text{No horizontal asymptotes}} \cr & \cr & {\text{Graph}} \cr} $$