Chapter 3 - Applications of Differentiation - 3.3 Exercises - Page 183: 47

\[\begin{align} & \text{The function is increasing on: } \\ & \left( 0,\frac{\pi }{2} \right),\left( \frac{7\pi }{6},\frac{3\pi }{2} \right),\left( \frac{11\pi }{6},2\pi \right) \\ & \text{The function is decreasing on: } \\ & \left( \frac{\pi }{2},\frac{7\pi }{6} \right),\left( \frac{3\pi }{2},\frac{11\pi }{6} \right) \\ \end{align}\]

\[\begin{align} & f\left( x \right)={{\sin }^{2}}x+\sin x \\ & \left( \text{a} \right)\text{Calculating the first derivative} \\ & f'\left( x \right)=\frac{d}{dx}\left[ {{\sin }^{2}}x+\sin x \right] \\ & f'\left( x \right)=2\sin x\cos x+\cos x \\ & \text{Calculating the critical points}\text{, set }f'\left( x \right)=0 \\ & 2\sin x\cos x+\cos x=0 \\ & \cos x\left( 2\sin x+1 \right)=0 \\ & \cos x=0,\text{ }2\sin x+1=0 \\ & \text{Solving for the interval }\left( 0,2\pi \right)\text{ we obtain the critical numbers} \\ & x=\frac{\pi }{2},\text{ }x=\frac{7\pi }{6},\text{ }x=\frac{3\pi }{2},\text{ }x=\frac{11\pi }{6} \\ & \text{Set the intervals:} \\ & \left( 0,\frac{\pi }{2} \right),\left( \frac{\pi }{2},\frac{7\pi }{6} \right),\left( \frac{7\pi }{6},\frac{3\pi }{2} \right),\left( \frac{3\pi }{2},\frac{11\pi }{6} \right),\left( \frac{11\pi }{6},2\pi \right) \\ & \\ & \text{Making a table of values }\\ & \begin{matrix} \text{Interval} & \text{Test Value} & \text{Sign of }f'\left( x \right) & \text{Conclusion} \\ \left( 0,\frac{\pi }{2} \right) & \frac{\pi }{4} & >0 & \text{Increasing} \\ \left( \frac{\pi }{2},\frac{7\pi }{6} \right) & \frac{5\pi }{6} & <0 & \text{Decreasing} \\ \left( \frac{7\pi }{6},\frac{3\pi }{2} \right) & \frac{4\pi }{3} & >0 & \text{Increasing} \\ \left( \frac{3\pi }{2},\frac{11\pi }{6} \right) & \frac{5\pi }{3} & <0 & \text{Decreasing} \\ \left( \frac{11\pi }{6},2\pi \right) & \frac{23\pi }{12} & >0 & \text{Increasing} \\ \end{matrix} \\ & \\ & \text{The function is increasing on: } \\ & \left( 0,\frac{\pi }{2} \right),\left( \frac{7\pi }{6},\frac{3\pi }{2} \right),\left( \frac{11\pi }{6},2\pi \right) \\ & \text{The function is decreasing on: } \\ & \left( \frac{\pi }{2},\frac{7\pi }{6} \right),\left( \frac{3\pi }{2},\frac{11\pi }{6} \right) \\ & \\ & *f'\left( x \right)\text{ changes from positive to negative at }x=\frac{\pi }{2},\text{ } \\ & \text{then }f\left( x \right)\text{ has a relative maximum at }\left( \frac{\pi }{2},f\left( \frac{\pi }{2} \right) \right) \\ & f\left( \frac{\pi }{4} \right)=2 \\ & *f'\left( x \right)\text{ changes from negative to positive at }x=\frac{7\pi }{6},\text{ } \\ & \text{then }f\left( x \right)\text{ has a relative minimum at }\left( \frac{7\pi }{6},f\left( \frac{7\pi }{6} \right) \right) \\ & f\left( \frac{7\pi }{6} \right)=-\frac{1}{4} \\ & *f'\left( x \right)\text{ changes from positive to negative at }x=\frac{3\pi }{2},\text{ } \\ & \text{then }f\left( x \right)\text{ has a relative maximum at }\left( \frac{3\pi }{2},f\left( \frac{3\pi }{2} \right) \right) \\ & f\left( \frac{3\pi }{2} \right)=0 \\ & *f'\left( x \right)\text{ changes from negative to positive at }x=\frac{11\pi }{6}, \\ & \text{then }f\left( x \right)\text{has a relative minimum at }\left( \frac{11\pi }{6},f\left( \frac{11\pi }{6} \right) \right) \\ & f\left( \frac{11\pi }{6} \right)=-\frac{1}{4} \\ & \\ & \left( \text{c} \right)\left( \text{graph} \right) \\ \end{align}\]