Chapter 3 - The Derivative In Graphing And Applications - 3.3 Analysis Of Functions III: Rational Functions, Cusps, And Vertical Tangents - Exercises Set 3.3 - Page 215: 37

$$\eqalign{ & {\text{The critical points are: }}\left( {0,4} \right){\text{ and }}\left( {1,3} \right) \cr & {\text{The inflection points are: }}\left( {8,4} \right){\text{ and }}\left( {0,4} \right) \cr} $$

$$\eqalign{ & {\text{Let }}f\left( x \right) = \frac{{8 + x}}{{2 + \root 3 \of x }} = \frac{{{{\left( 2 \right)}^3} + {{\left( {{x^{1/3}}} \right)}^3}}}{{2 + \root 3 \of x }} \cr & {\text{Factoring the numerator, }} \cr & {\text{recall that }}{a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right) \cr & f\left( x \right) = \frac{{{{\left( 2 \right)}^3} + {{\left( {{x^{1/3}}} \right)}^3}}}{{2 + \root 3 \of x }} = \frac{{\left( {2 + \root 3 \of x } \right)\left( {{2^2} - 2{x^{1/3}} + {{\left( {{x^{1/3}}} \right)}^2}} \right)}}{{2 + \root 3 \of x }} \cr & {\text{Simplifying}} \cr & f\left( x \right) = \frac{{\left( {2 + \root 3 \of x } \right)\left( {4 - 2{x^{1/3}} + {x^{2/3}}} \right)}}{{2 + \root 3 \of x }} \cr & f\left( x \right) = 4 - 2{x^{1/3}} + {x^{2/3}} \cr & {\text{Differentiate}} \cr & f\left( x \right) = - \frac{2}{3}{x^{ - 2/3}} + \frac{2}{3}{x^{ - 1/3}} \cr & \cr & {\text{Calculate the critical points}} \cr & f'\left( x \right) = 0 \cr & - \frac{2}{3}{x^{ - 2/3}} + \frac{2}{3}{x^{ - 1/3}} = 0 \cr & - {x^{ - 2/3}} + {x^{ - 1/3}} = 0 \cr & - {x^{ - 2/3}} + {x^{ - 1/3}} = 0 \cr & {\text{Solving we obtain}} \cr & x = 1 \cr & {\text{The derivative is not defined at }}x = 0 \cr & {\text{Calculate }}f\left( 0 \right){\text{ and }}f\left( 1 \right) \cr & f\left( 0 \right) = 4 \cr & f\left( 1 \right) = {\text{3}} \cr & {\text{The critical points are: }}\left( {0,4} \right){\text{ and }}\left( {1,3} \right) \cr & \cr & {\text{Calculate the second derivative}} \cr & f''\left( x \right) = \frac{d}{{dx}}\left[ { - \frac{2}{3}{x^{ - 2/3}} + \frac{2}{3}{x^{ - 1/3}}} \right] \cr & f''\left( x \right) = - \frac{2}{3}\left( { - \frac{2}{3}} \right){x^{ - 5/3}} + \frac{2}{3}\left( { - \frac{1}{3}} \right){x^{ - 4/3}} \cr & f''\left( x \right) = \frac{4}{9}{x^{ - 5/3}} - \frac{2}{9}{x^{ - 4/3}} \cr & \frac{4}{9}{x^{ - 5/3}} - \frac{2}{9}{x^{ - 4/3}} = 0 \cr & {\text{Solving we obtain}} \cr & x = 8,{\text{ and it's not defined at }}x = 0 \cr & f\left( 8 \right) = 4 \cr & {\text{The inflection points are: }}\left( {8,4} \right){\text{ and }}\left( {0,4} \right) \cr} $$