Chapter 3 - Applications of Differentiation - 3.4 Exercises - Page 192: 27

$$\eqalign{ & {\text{No inflection points}} \cr & {\text{Concave downward}}:{\text{ }}\left( {\pi ,2\pi } \right){\text{ and }}\left( {3\pi ,4\pi } \right) \cr & {\text{Concave upward}}:{\text{ }}\left( {0,\pi } \right){\text{ and }}\left( {2\pi ,3\pi } \right) \cr} $$

$$\eqalign{ & f\left( x \right) = \sec \left( {x - \frac{\pi }{2}} \right),{\text{ }}\left( {0,4\pi } \right) \cr & {\text{The function is not continuous at }}x = \pi ,{\text{ }}x = 2\pi {\text{ and }}x = 3\pi \cr & {\text{on the interval }}\left( {0,4\pi } \right) \cr & {\text{Calculate the first and second derivatives}} \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {\sec \left( {x - \frac{\pi }{2}} \right)} \right] \cr & f'\left( x \right) = \sec \left( {x - \frac{\pi }{2}} \right)\tan \left( {x - \frac{\pi }{2}} \right)\left( 1 \right) \cr & f''\left( x \right) = \frac{d}{{dx}}\left[ {\sec \left( {x - \frac{\pi }{2}} \right)\tan \left( {x - \frac{\pi }{2}} \right)} \right] \cr & f''\left( x \right) = \sec \left( {x - \frac{\pi }{2}} \right){\sec ^2}\left( {x - \frac{\pi }{2}} \right) + {\tan ^2}\left( {x - \frac{\pi }{2}} \right)\sec \left( {x - \frac{\pi }{2}} \right) \cr & f''\left( x \right) = {\sec ^3}\left( {x - \frac{\pi }{2}} \right) + {\tan ^2}\left( {x - \frac{\pi }{2}} \right)\sec \left( {x - \frac{\pi }{2}} \right) \cr & {\text{Factoring}} \cr & f''\left( x \right) = \sec \left( {x - \frac{\pi }{2}} \right)\left[ {{{\sec }^2}\left( {x - \frac{\pi }{2}} \right) + {{\tan }^2}\left( {x - \frac{\pi }{2}} \right)} \right] \cr & {\text{The function is not continuous for the domain of }}f\left( x \right),{\text{ at}} \cr & x = \pi ,{\text{ }}x = 2\pi {\text{ and }}x = 3\pi \cr & {\text{We obtain the intervals}} \cr & \left( {0,\pi } \right),\left( {\pi ,2\pi } \right),\left( {2\pi ,3\pi } \right),\left( {3\pi ,4\pi } \right) \cr & {\text{Making a table of values }}\left( {{\text{See examples on page 188 }}} \right) \cr} $$ \[\boxed{\begin{array}{*{20}{c}} {{\text{Interval}}}&{{\text{Test Value}}}&{{\text{Sign of }}f''\left( x \right)}&{{\text{Conclusion}}} \\ {\left( {0,\pi } \right)}&{x = \frac{{2\pi }}{3}}& + &{{\text{C}}{\text{. upward}}} \\ {\left( {\pi ,2\pi } \right)}&{x = \frac{{4\pi }}{3}}& - &{{\text{C}}{\text{. downward}}} \\ {\left( {2\pi ,3\pi } \right)}&{x = \frac{{7\pi }}{3}}& + &{{\text{C}}{\text{. upward}}} \\ {\left( {3\pi ,4\pi } \right)}&{x = \frac{{10\pi }}{3}}& - &{{\text{C}}{\text{. downward}}} \end{array}}\] $$\eqalign{ & {\text{Summary:}} \cr & {\text{There are no values at which }}f''\left( x \right) = 0 \cr & {\text{No inflection points}} \cr & {\text{Concave downward}}:{\text{ }}\left( {\pi ,2\pi } \right){\text{ and }}\left( {3\pi ,4\pi } \right) \cr & {\text{Concave upward}}:{\text{ }}\left( {0,\pi } \right){\text{ and }}\left( {2\pi ,3\pi } \right) \cr & {\text{The following graph confirms the result:}} \cr} $$