Chapter 3 - Applications of Differentiation - 3.5 Exercises - Page 204: 80

$$\eqalign{ & {\text{Inflection point }}\left( {0,0} \right) \cr & {\text{Vertical asymptotes: none}} \cr & {\text{Horizontal asymptotes at }}y = - \frac{2}{{\sqrt 3 }}{\text{ and }}y = \frac{2}{{\sqrt 3 }} \cr & {\text{No relative extrema}} \cr & {\text{Domain}}:\left( { - \infty ,\infty } \right) \cr} $$

$$\eqalign{ & f\left( x \right) = \frac{{2x}}{{\sqrt {3{x^2} + 1} }} \cr & 3{x^2} + 1{\text{ is always positive and }} > {\text{0, then the domain of the }} \cr & {\text{function is }}D:\left( { - \infty ,\infty } \right) \cr & \cr & {\text{*Differentiating}} \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {\frac{{2x}}{{\sqrt {3{x^2} + 1} }}} \right] \cr & f'\left( x \right) = \frac{{2\sqrt {3{x^2} + 1} - 2x\left( {\frac{{6x}}{{2\sqrt {3{x^2} + 1} }}} \right)}}{{3{x^2} + 1}} \cr & f'\left( x \right) = \frac{{4\left( {3{x^2} + 1} \right) - 2x\left( {6x} \right)}}{{2{{\left( {3{x^2} + 1} \right)}^{3/2}}}} \cr & f'\left( x \right) = \frac{{12{x^2} + 4 - 12{x^2}}}{{2{{\left( {3{x^2} + 1} \right)}^{3/2}}}} \cr & f'\left( x \right) = \frac{4}{{2{{\left( {3{x^2} + 1} \right)}^{3/2}}}} \cr & f'\left( x \right) = \frac{2}{{{{\left( {3{x^2} + 1} \right)}^{3/2}}}} \cr & {\text{Set }}f'\left( x \right) = 0 \cr & \frac{2}{{{{\left( {3{x^2} + 1} \right)}^{3/2}}}} = 0 \cr & {\text{There are no solution}}{\text{.}} \cr & {\text{Then there are no relative extrema}}{\text{.}} \cr & \cr & {\text{*Find the second derivative}} \cr & f''\left( x \right) = \frac{d}{{dx}}\left[ {\frac{2}{{{{\left( {3{x^2} + 1} \right)}^{3/2}}}}} \right] \cr & {\text{Using a CAS we obtain}} \cr & f''\left( x \right) = - 3{\left( {4{x^2} + 1} \right)^{ - 1/2}}\left( {6x} \right) \cr & f''\left( x \right) = - \frac{{18x}}{{\sqrt {4{x^2} + 1} }} \cr & {\text{Set }}f''\left( x \right) = 0 \cr & 18x = 0 \cr & x = 0 \cr & {\text{For }}\left( { - \infty ,0} \right) \to f''\left( { - 1} \right) > 0 \cr & {\text{For }}\left( {0,\infty } \right) \to f''\left( 1 \right) < 0 \cr & {\text{Then there is an inflection point at }}x = 0 \cr & f\left( 0 \right) = 0 \to {\text{Inflection point }}\left( {0,0} \right) \cr & \cr & {\text{There are no vertical asymptotes because the denominator}} \cr & {\text{is never 0}}{\text{.}} \cr & \cr & *{\text{Find the horizontal asymptotes}} \cr & \mathop {\lim }\limits_{x \to \infty } f\left( x \right) = \mathop {\lim }\limits_{x \to \infty } \frac{{2x}}{{\sqrt {3{x^2} + 1} }} = \frac{2}{{\sqrt 3 }} \cr & \mathop {\lim }\limits_{x \to - \infty } f\left( x \right) = \mathop {\lim }\limits_{x \to - \infty } \frac{{2x}}{{\sqrt {3{x^2} + 1} }} = - \frac{2}{{\sqrt 3 }} \cr & {\text{Horizontal asymptotes at }}y = - \frac{2}{{\sqrt 3 }}{\text{ and }}y = \frac{2}{{\sqrt 3 }} \cr & \cr \cr} $$