Chapter 3 - Applications of Differentiation - 3.6 Exercises - Page 212: 27

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

$$\eqalign{ & f\left( x \right) = \frac{{ - 2x}}{{\sqrt {{x^2} + 7} }} \cr & {x^2} + 7{\text{ is always positive and }} > {\text{0, then the domain of the }} \cr & {\text{function is }}D:\left( { - \infty ,\infty } \right) \cr & \cr & {\text{*Differentiating by using technology we obtain:}} \cr & f'\left( x \right) = - \frac{{14}}{{{{\left( {{x^2} + 7} \right)}^{3/2}}}} \cr & {\text{Set }}f'\left( x \right) = 0 \cr & - \frac{{14}}{{{{\left( {{x^2} + 7} \right)}^{3/2}}}} = 0 \cr & {\text{There are no solution}}{\text{.}} \cr & {\text{Then there are no relative extrema}}{\text{.}} \cr & \cr & {\text{*Find the second derivative}} \cr & f''\left( x \right) = \frac{d}{{dx}}\left[ { - \frac{{14}}{{{{\left( {{x^2} + 7} \right)}^{3/2}}}}} \right] \cr & f''\left( x \right) = \frac{{42x}}{{{{\left( {{x^2} + 7} \right)}^{5/2}}}} \cr & {\text{Set }}f''\left( x \right) = 0 \cr & 42x = 0 \cr & x = 0 \cr & {\text{For }}\left( { - \infty ,0} \right) \to f''\left( { - 1} \right) > 0 \cr & {\text{For }}\left( {0,\infty } \right) \to f''\left( 1 \right) < 0 \cr & {\text{Then there is an inflection point at }}x = 0 \cr & f\left( 0 \right) = 0 \to {\text{Inflection point }}\left( {0,0} \right) \cr & \cr & {\text{There are no vertical asymptotes because the denominator}} \cr & {\text{is never 0}}{\text{.}} \cr & \cr & *{\text{Find the horizontal asymptotes using the graph}} \cr & \mathop {\lim }\limits_{x \to \infty } f\left( x \right) = - 2 \cr & \mathop {\lim }\limits_{x \to - \infty } f\left( x \right) = 2 \cr & {\text{Horizontal asymptotes at }}y = - 2{\text{ and }}y = 2 \cr & \cr & *{\text{From the graph we can notice that the function}} \cr & {\text{has symmetry with respect to the origin}}{\text{.}} \cr & \cr } $$