Chapter 3 - Applications of Differentiation - 3.6 Exercises - Page 212: 28

$$\eqalign{ & {\text{Inflection point }}\left( {0,0} \right) \cr } $$

$$\eqalign{ & f\left( x \right) = \frac{{4x}}{{\sqrt {{x^2} + 15} }} \cr & {x^2} + 15{\text{ is always positive and }} > {\text{0, then the domain of the }} \cr & {\text{function is }}D:\left( { - \infty ,\infty } \right) \cr & \cr & {\text{*Differentiating by using technology we obtain:}} \cr & f'\left( x \right) = \frac{{60}}{{{{\left( {{x^2} + 15} \right)}^{3/2}}}} \cr & {\text{Set }}f'\left( x \right) = 0 \cr & \frac{{60}}{{{{\left( {{x^2} + 15} \right)}^{3/2}}}} = 0 \cr & {\text{There are no solution}}{\text{.}} \cr & {\text{Then there are no relative extrema}}{\text{.}} \cr & \cr & {\text{*Find the second derivative}} \cr & f''\left( x \right) = \frac{d}{{dx}}\left[ {\frac{{60}}{{{{\left( {{x^2} + 15} \right)}^{3/2}}}}} \right] \cr & f''\left( x \right) = - \frac{{180x}}{{{{\left( {{x^2} + 15} \right)}^{5/2}}}} \cr & {\text{Set }}f''\left( x \right) = 0 \cr & 180x = 0 \cr & x = 0 \cr & {\text{For }}\left( { - \infty ,0} \right) \to f''\left( { - 1} \right) > 0 \cr & {\text{For }}\left( {0,\infty } \right) \to f''\left( 1 \right) < 0 \cr & {\text{Then there is an inflection point at }}x = 0 \cr & f\left( 0 \right) = 0 \to {\text{Inflection point }}\left( {0,0} \right) \cr & \cr & {\text{There are no vertical asymptotes because the denominator}} \cr & {\text{is never 0}}{\text{.}} \cr & \cr & *{\text{Find the horizontal asymptotes using the graph}} \cr & \mathop {\lim }\limits_{x \to \infty } f\left( x \right) = - 4 \cr & \mathop {\lim }\limits_{x \to - \infty } f\left( x \right) = 4 \cr & {\text{Horizontal asymptotes at }}y = - 4{\text{ and }}y = 4 \cr & \cr & *{\text{From the graph we can notice that the function}} \cr & {\text{has symmetry with respect to the origin}}{\text{.}} \cr & \cr } $$