Chapter 3 - Applications of Differentiation - 3.5 Exercises - Page 203: 72

Graph

$$\eqalign{ & y = \frac{4}{{{x^2}}} + 1 \cr & {\text{Find the }}y{\text{ intercept, let }}x = 0 \cr & y = \frac{4}{{{{\left( 0 \right)}^2}}} + 1 \cr & {\text{No }}y{\text{ - intercept}}{\text{.}} \cr & {\text{Find the }}x{\text{ intercept, let }}y = 0 \cr & 0 = \frac{4}{{{x^2}}} + 1 \cr & \frac{4}{{{x^2}}} = - 1 \cr & {x^2} = - \frac{1}{4} \cr & {\text{No }}x{\text{ - intercept}}{\text{.}} \cr & \cr & *{\text{Find the extrema}} \cr & {\text{Differentiate}} \cr & y' = \frac{d}{{dx}}\left[ {\frac{4}{{{x^2}}} + 1} \right] \cr & y' = 4\left( { - 2{x^{ - 3}}} \right) \cr & y' = - \frac{8}{{{x^3}}} \cr & - \frac{8}{{{x^3}}},{\text{ there are no values at which }}y' = 0. \cr & {\text{No relative extrema}}{\text{.}} \cr & \cr & {\text{*Calculate the asymptotes}} \cr & \frac{4}{{{x^2}}} + 1 \cr & {x^2} = 0 \to x = 0 \cr & {\text{Vertical asymptote at }}x = 0 \cr & \mathop {\lim }\limits_{x \to \infty } \left( {\frac{4}{{{x^2}}} + 1} \right) = 1 \cr & \mathop {\lim }\limits_{x \to - \infty } \left( {\frac{4}{{{x^2}}} + 1} \right) = 1 \cr & {\text{Horizontal asymptote }}y = 1 \cr & \cr & {\text{*Symmetry}} \cr & f\left( { - x} \right) = \frac{4}{{{{\left( { - x} \right)}^2}}} + 1 \cr & f\left( { - x} \right) = \frac{4}{{{x^2}}} + 1 \cr & f\left( { - x} \right) = f\left( x \right) \cr & f\left( { - x} \right) = f\left( x \right){\text{ The function is even}} \cr & {\text{Symmetry about the }}y{\text{ - axis}} \cr & \cr & {\text{Graph}} \cr} $$