Chapter 4 - Applications of the Derivative - 4.3 Graphing Functions - 4.3 Exercises - Page 268: 23

$$\eqalign{ & {\text{Domain}}:\left[ { - 2\pi ,2\pi } \right] \cr & {\text{Relative maximum at:}} \cr & \left( { - \frac{{11\pi }}{6}, - \frac{{11\pi }}{6} + \sqrt 3 } \right){\text{ and }}\left( {\frac{\pi }{6},\frac{\pi }{6} + \sqrt 3 } \right) \cr & {\text{Relative minimum at:}} \cr & \left( { - \frac{{7\pi }}{6}, - \frac{{7\pi }}{6} - \sqrt 3 } \right),\left( {\frac{{5\pi }}{6}, - \frac{{7\pi }}{6} - \sqrt 3 } \right) \cr & {\text{Inflection points at: }} \cr & x = - \frac{{3\pi }}{2},{\text{ }}x = - \frac{\pi }{2},{\text{ }}x = \frac{\pi }{2},{\text{ }}x = \frac{{3\pi }}{2} \cr & {\text{Vertical asymptotes: none}} \cr & {\text{Horizontal asymptotes}}:{\text{none}} \cr} $$

$$\eqalign{ & f\left( x \right) = x + 2\cos x{\text{ on }}\left[ { - 2\pi ,2\pi } \right] \cr & {\text{The domain of the function is }}D:\left[ { - 2\pi ,2\pi } \right] \cr & \cr & {\text{*Differentiating }} \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {x + 2\cos x} \right] \cr & f'\left( x \right) = 1 - 2\sin x \cr & {\text{Set }}f'\left( x \right) = 0{\text{ to find the critical points}} \cr & 1 - 2\sin x = 0 \cr & \sin x = \frac{1}{2} \cr & {\text{On the interval }}\left[ { - 2\pi ,2\pi } \right],{\text{ }}\sin x = \frac{1}{2}{\text{ at: }} \cr & x = - \frac{{11\pi }}{6},{\text{ }}x = - \frac{{7\pi }}{6},{\text{ }}x = \frac{\pi }{6},{\text{ }}x = \frac{{5\pi }}{6} \cr & \cr & {\text{*Find the second derivative}} \cr & f''\left( x \right) = \frac{d}{{dx}}\left[ {1 - 2\sin x} \right] \cr & f''\left( x \right) = - 2\cos x \cr & {\text{Evaluate }}f''\left( x \right){\text{ at the critical points}} \cr & *f''\left( { - \frac{{11\pi }}{6}} \right) = - \sqrt 3 < 0,{\text{ relative maximum}} \cr & f\left( { - \frac{{11\pi }}{6}} \right) = - \frac{{11\pi }}{6} + \sqrt 3 ,{\text{ }}\left( { - \frac{{11\pi }}{6}, - \frac{{11\pi }}{6} + \sqrt 3 } \right) \cr & *f''\left( { - \frac{{7\pi }}{6}} \right) = \sqrt 3 > 0,{\text{ relative minimum}} \cr & f\left( { - \frac{{7\pi }}{6}} \right) = - \frac{{7\pi }}{6} - \sqrt 3 ,{\text{ }}\left( { - \frac{{7\pi }}{6}, - \frac{{7\pi }}{6} - \sqrt 3 } \right) \cr & *f''\left( {\frac{\pi }{6}} \right) = - \sqrt 3 < 0,{\text{ relative maximum}} \cr & f\left( {\frac{\pi }{6}} \right) = \frac{\pi }{6} + \sqrt 3 ,{\text{ }}\left( {\frac{\pi }{6},\frac{\pi }{6} + \sqrt 3 } \right) \cr & *f''\left( {\frac{{5\pi }}{6}} \right) = \sqrt 3 > 0,{\text{ relative minimum}} \cr & f\left( {\frac{{5\pi }}{6}} \right) = \frac{{5\pi }}{6} - \sqrt 3 ,{\text{ }}\left( {\frac{{5\pi }}{6}, - \frac{{7\pi }}{6} - \sqrt 3 } \right) \cr & \cr & *{\text{Find the Inflection points}}{\text{, set }}f''\left( x \right) = 0 \cr & - 2\cos x = 0 \cr & {\text{On the interval }}\left[ { - 2\pi ,2\pi } \right],{\text{ }} - 2\cos x = 0{\text{ at: }} \cr & x = - \frac{{3\pi }}{2},{\text{ }}x = - \frac{\pi }{2},{\text{ }}x = \frac{\pi }{2},{\text{ }}x = \frac{{3\pi }}{2} \cr & \cr & {\text{*There are no vertical asymptotes because the denominator}} \cr & {\text{is never 0}}{\text{.}} \cr & *{\text{There are no horizontal asymptotes}} \cr & \cr & {\text{Summary}} \cr & {\text{Domain}}:\left[ { - 2\pi ,2\pi } \right] \cr & {\text{Relative maximum at:}} \cr & \left( { - \frac{{11\pi }}{6}, - \frac{{11\pi }}{6} + \sqrt 3 } \right){\text{ and }}\left( {\frac{\pi }{6},\frac{\pi }{6} + \sqrt 3 } \right) \cr & {\text{Relative minimum at:}} \cr & \left( { - \frac{{7\pi }}{6}, - \frac{{7\pi }}{6} - \sqrt 3 } \right),\left( {\frac{{5\pi }}{6}, - \frac{{7\pi }}{6} - \sqrt 3 } \right) \cr & {\text{Inflection points at: }} \cr & x = - \frac{{3\pi }}{2},{\text{ }}x = - \frac{\pi }{2},{\text{ }}x = \frac{\pi }{2},{\text{ }}x = \frac{{3\pi }}{2} \cr & {\text{Vertical asymptotes: none}} \cr & {\text{Horizontal asymptotes}}:{\text{none}} \cr & \cr & {\text{Graph}} \cr} $$