Chapter 5 - Logarithmic, Exponential, and Other Transcendental Functions - 5.4 Exercises - Page 353: 75

$$\eqalign{ & {\text{Relative minimum at }}\left( {0,0} \right) \cr & {\text{Relative maximum at }}\left( {2,4{e^{ - 2}}} \right) \cr & {\text{Inflection points at: }}\left( {3.414,0.3835} \right){\text{ and}}\left( {0.586,0.19101} \right) \cr} $$

$$\eqalign{ & f\left( x \right) = {x^2}{e^{ - x}} \cr & {\text{Calculate the first derivative}} \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {{x^2}{e^{ - x}}} \right] \cr & f'\left( x \right) = - {x^2}{e^{ - x}} + 2x{e^{ - x}} \cr & {\text{Set }}f'\left( x \right) = 0 \cr & - {x^2}{e^{ - x}} + 2x{e^{ - x}} = 0 \cr & - x{e^{ - x}}\left( {x - 2} \right) = 0 \cr & x = 0,{\text{ }}x = 2 \cr & \cr & *{\text{Calculate the second derivative}} \cr & f''\left( x \right) = \frac{d}{{dx}}\left[ { - {x^2}{e^{ - x}} + 2x{e^{ - x}}} \right] \cr & f''\left( x \right) = - 2x{e^{ - x}} + {x^2}{e^{ - x}} + 2{e^{ - x}} - 2x{e^{ - x}} \cr & f''\left( x \right) = {x^2}{e^{ - x}} + 2{e^{ - x}} - 4x{e^{ - x}} \cr & f''\left( x \right) = \left( {{x^2} - 4x + 2} \right){e^{ - x}} \cr & \cr & {\text{Evaluate the second derivative at }}x = 0{\text{ and }}x = 2 \cr & f''\left( 0 \right) = \left( 2 \right){e^{ - 2}} > 0 \cr & f''\left( 2 \right) = \left( {4 - 8 + 2} \right) < 0 \cr & {\text{Then by the second derivative test }} \cr & f\left( x \right){\text{ has a relative minimum at }}\left( {0,f\left( 0 \right)} \right) \cr & f\left( x \right){\text{ has a relative maximum at }}\left( {2,f\left( 2 \right)} \right) \cr & f\left( 0 \right) = {\left( 0 \right)^2}{e^{ - 0}} \to {\text{Relative minimum at }}\left( {0,0} \right) \cr & f\left( 2 \right) = {\left( 2 \right)^2}{e^{ - 2}} = 4{e^{ - 2}} \to {\text{Relative maximum at }}\left( {2,4{e^{ - 2}}} \right) \cr & \cr & {\text{Set }}f''\left( x \right) = 0 \cr & {x^2} - 4x + 2 = 0 \cr & {\text{By the quadratic formula}} \cr & {x_1} = 2 + \sqrt 2 ,{\text{ }}{x_2} = 2 - \sqrt 2 \cr & \cr & {\text{Inflection points at:}} \cr & \left( {2 + \sqrt 2 ,f\left( {2 + \sqrt 2 } \right)} \right){\text{ and }}\left( {2 - \sqrt 2 ,f\left( {2 - \sqrt 2 } \right)} \right) \cr & f\left( {2 + \sqrt 2 } \right) = {\left( {2 + \sqrt 2 } \right)^2}{e^{ - \left( {2 + \sqrt 2 } \right)}} \approx 0.3835 \cr & f\left( {2 - \sqrt 2 } \right) = {\left( {2 - \sqrt 2 } \right)^2}{e^{ - \left( {2 - \sqrt 2 } \right)}} = 0.19101 \cr & {\text{Inflection points at:}} \cr & \left( {3.414,0.3835} \right){\text{ and }}\left( {0.586,0.19101} \right) \cr & \cr & {\text{Graph}} \cr} $$