Chapter 3 - Applications of Differentiation - Review Exercises - Page 238: 33

$$\eqalign{ & \left( {\text{a}} \right){\text{ }}x = \frac{{3\pi }}{4}{\text{ and }}x = \frac{{7\pi }}{4} \cr & \left( {\text{b}} \right){\text{Increasing: }}\left( {\frac{{3\pi }}{4},\frac{{7\pi }}{4}} \right),{\text{ Decreasing on }}\left( {0,\frac{{3\pi }}{4}} \right),\left( {\frac{{7\pi }}{4},2\pi } \right) \cr & \left( {\text{c}} \right){\text{Relative maximum at }}\left( {\frac{{7\pi }}{4},\sqrt 2 } \right) \cr & {\text{Relative minumum at }}\left( {\frac{{3\pi }}{4}, - \sqrt 2 } \right) \cr} $$

$$\eqalign{ & f\left( x \right) = \cos x - \sin x,{\text{ }}\left( {0,2\pi } \right) \cr & \cr & \left( {\text{a}} \right){\text{Calculating the first derivative}} \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {\cos x - \sin x} \right] \cr & f'\left( x \right) = - \sin x - \cos x \cr & {\text{Calculating the critical points, set }}f'\left( x \right) = 0 \cr & f'\left( x \right) = 0 \cr & - \sin x - \cos x = 0 \cr & - \sin x = \cos x \cr & - \frac{{\sin x}}{{\cos x}} = 1 \cr & \tan x = - 1 \cr & {\text{On the interval }}\left( {0,2\pi } \right){\text{ }}\tan x = - 1{\text{ for }}x = \frac{{3\pi }}{4},\frac{{7\pi }}{4} \cr & \cr & \left( {\text{b}} \right){\text{Set the intervals }}\left( {0,\frac{{3\pi }}{4}} \right),\left( {\frac{{3\pi }}{4},\frac{{7\pi }}{4}} \right),\left( {\frac{{7\pi }}{4},2\pi } \right) \cr & {\text{Making a table of values }}\left( {{\text{See examples on page 180 }}} \right) \cr} $$ \[\boxed{\begin{array}{*{20}{c}} {{\text{Interval}}}&{{\text{Test Value}}}&{{\text{Sign of }}f'\left( x \right)}&{{\text{Conclusion}}} \\ {\left( {0,\frac{{3\pi }}{4}} \right)}&{x = \frac{\pi }{2}}&{ - 1 < 0}&{{\text{Decreasing}}} \\ {\left( {\frac{{3\pi }}{4},\frac{{7\pi }}{4}} \right)}&{x = \pi }&{1 > 0}&{{\text{Increasing}}} \\ {\left( {\frac{{7\pi }}{4},2\pi } \right)}&{x = \frac{{11\pi }}{6}}&{ \approx - 0.366 < 0}&{{\text{Decreasing}}} \end{array}}\] $$\eqalign{ & \left( {\text{c}} \right){\text{ By Theorem 3}}{\text{.6}} \cr & *f'\left( x \right){\text{changes from negative to positive at }}x = \frac{{3\pi }}{4},{\text{so }}f\left( x \right) \cr & {\text{has a relative minimum at }}\left( {\frac{{3\pi }}{4},f\left( {\frac{{3\pi }}{4}} \right)} \right) \cr & f\left( {\frac{{3\pi }}{4}} \right) = \cos \left( {\frac{{3\pi }}{4}} \right) - \sin \left( {\frac{{3\pi }}{4}} \right) = - \sqrt 2 \cr & {\text{Relative minumum at }}\left( {\frac{{3\pi }}{4}, - \sqrt 2 } \right) \cr & *f'\left( x \right){\text{changes from positive to negative at }}x = \frac{{7\pi }}{4},{\text{so }}f\left( x \right) \cr & {\text{has a relative maximum at }}\left( {\frac{{7\pi }}{4},f\left( {\frac{{7\pi }}{4}} \right)} \right) \cr & f\left( {\frac{{7\pi }}{4}} \right) = \cos \left( {\frac{{7\pi }}{4}} \right) - \sin \left( {\frac{{7\pi }}{4}} \right) = \sqrt 2 \cr & {\text{Relative maximum at }}\left( {\frac{{7\pi }}{4},\sqrt 2 } \right) \cr & \cr & \left( {\text{d}} \right){\text{Graph}} \cr} $$