Chapter 3 - Applications of Differentiation - 3.6 Exercises - Page 212: 33

$$\eqalign{ & {\text{*No }}x{\text{ - intercepts}} \cr & {\text{*No }}y{\text{ - intercepts}} \cr & {\text{Domain}}:\left( {0,\frac{\pi }{2}} \right) \cr & {\text{Relative minimum at }}\left( {\frac{\pi }{4},4\sqrt 2 } \right) \cr & {\text{Inflection points: none}} \cr & {\text{Vertical asymptotes: }}x = 0,\,x = \frac{\pi }{2} \cr & {\text{Horizontal asymptotes}}:{\text{none}} \cr & {\text{No symmetry}} \cr} $$

$$\eqalign{ & f\left( x \right) = 2\left( {\csc x + \sec x} \right),{\text{ }}0 < x < \frac{\pi }{2} \cr & {\text{From the graph we can notice that:}} \cr & {\text{*No }}x{\text{ - intercepts}} \cr & {\text{*No }}y{\text{ - intercepts}} \cr & \cr & {\text{The domain of the function is }}D:\left( {0,\frac{\pi }{2}} \right) \cr & \cr & {\text{*Differentiating }} \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {2\left( {\csc x + \sec x} \right)} \right] \cr & f'\left( x \right) = 2\left( { - \csc x\cot x + \sec x\tan x} \right) \cr & {\text{Set }}f'\left( x \right) = 0 \cr & 2\left( { - \csc x\cot x + \sec x\tan x} \right) = 0 \cr & {\text{For the interval }}\left( {0,\frac{\pi }{2}} \right){\text{ we obatin }}x = \frac{\pi }{4} \cr & {\text{From the graph:}} \cr & f'\left( {\frac{\pi }{8}} \right) < 0{\text{ and }}f'\left( {\frac{{3\pi }}{8}} \right) > 0 \cr & {\text{Then by the first derivative test, there are a relative}} \cr & {\text{minimum at }}x = \frac{\pi }{4} \cr & f\left( {\frac{\pi }{4}} \right) = 2\left( {\csc \frac{\pi }{4} + \sec \frac{\pi }{4}} \right) = 2\left( {\sqrt 2 + \sqrt 2 } \right) = 4\sqrt 2 \cr & \to {\text{Relative minimum at }}\left( {\frac{\pi }{4},4\sqrt 2 } \right) \cr & \cr & {\text{From the graph:}} \cr & {\text{*There are no inflection points}} \cr & *{\text{There are no horizontal asymptotes}} \cr & {\text{Vertical asymptotes: }}x = 0,\,x = \frac{\pi }{2} \cr & \cr } $$