Chapter 3 - Applications of Differentiation - Review Exercises - Page 239: 73

$$\eqalign{ & y{\text{ - intercept }}\left( {0, - \frac{5}{2}} \right) \cr & x{\text{ - intercept: }}\left( {\frac{5}{3},0} \right) \cr & {\text{No relative extrema}} \cr & {\text{No inflection points}} \cr & {\text{Vertical asymptote at }}x = 2 \cr & {\text{Horizontal asymptote }}y = - 3 \cr} $$

$$\eqalign{ & f\left( x \right) = \frac{{5 - 3x}}{{x - 2}} \cr & {\text{Find the }}y{\text{ intercept, let }}x = 0 \cr & y = \frac{{5 - 3\left( 0 \right)}}{{\left( 0 \right) - 2}} \cr & y = - \frac{5}{2} \cr & y{\text{ - intercept }}\left( {0, - \frac{5}{2}} \right) \cr & {\text{Find the }}x{\text{ intercept, let }}y = 0 \cr & \frac{{5 - 3x}}{{x - 2}} = 0 \cr & x = \frac{5}{3} \cr & x{\text{ - intercept }}\left( {\frac{5}{3},0} \right) \cr & \cr & *{\text{Find the extrema}} \cr & {\text{Differentiate}} \cr & y' = \frac{d}{{dx}}\left[ {\frac{{5 - 3x}}{{x - 2}}} \right] \cr & y' = \frac{{\left( {x - 2} \right)\left( { - 3} \right) - \left( {5 - 3x} \right)}}{{{{\left( {x - 2} \right)}^2}}} \cr & y' = \frac{{ - 3x + 6 - 5 + 3x}}{{{{\left( {x - 2} \right)}^2}}} \cr & y' = \frac{1}{{{{\left( {x - 2} \right)}^2}}} \cr & {\text{Let }}y' = 0 \cr & \frac{1}{{{{\left( {x - 2} \right)}^2}}} = 0,{\text{ there are no values at which }}y' = 0, \cr & {\text{No relative extrema}}{\text{.}} \cr & \cr & *{\text{Find the Inflection points}} \cr & y'' = \frac{d}{{dx}}\left[ {\frac{1}{{{{\left( {x - 2} \right)}^2}}}} \right] \cr & y'' = - \left( { - 2} \right){\left( {x - 2} \right)^{ - 3}} \cr & y'' = \frac{2}{{{{\left( {x - 2} \right)}^3}}} \cr & {\text{Let }}y'' = 0 \cr & \frac{2}{{{{\left( {x - 2} \right)}^3}}} = 0 \cr & {\text{There are no values at which }}y'' = 0. \cr & {\text{No inflection points}}{\text{.}} \cr & \cr & {\text{*Calculate the asymptotes}} \cr & \frac{{5 - 3x}}{{x - 2}} \cr & x - 2 = 0 \to x = 2 \cr & {\text{Vertical asymptote at }}x = 2 \cr & \mathop {\lim }\limits_{x \to \infty } \left( {\frac{{5 - 3x}}{{x - 2}}} \right) = - 3 \cr & \mathop {\lim }\limits_{x \to - \infty } \left( {\frac{{5 - 3x}}{{x - 2}}} \right) = - 3 \cr & {\text{Horizontal asymptote }}y = - 3 \cr & \cr } $$