Chapter 4 - Applications of the Derivative - Review Exercises - Page 330: 6

\[\begin{align} & \text{Absolute maximum: }\left( -\frac{3\pi }{4},4 \right)\text{ and }\left( \frac{\pi }{4},4 \right) \\ & \text{Absolute minimum: }\left( -\frac{\pi }{4},2 \right)\text{ and }\left( \frac{3\pi }{4},2 \right)\text{ } \\ \end{align}\]

\[\begin{align} & f\left( x \right)=\sin 2x+3\text{ on }\left[ -\pi ,\pi \right] \\ & \text{Differentiate} \\ & f'\left( x \right)=\sin 2x+3 \\ & f'\left( x \right)=\cos 2x\left( 2 \right)+0 \\ & f'\left( x \right)=2\cos 2x \\ & \text{Calculate the critial points, set }f'\left( x \right)=0 \\ & 2\cos 2x=0 \\ & \cos 2x=0 \\ & \text{For the interval }\left[ -\pi ,\pi \right]\text{ }\cos 2x=0,\text{ when } \\ & x=\pm \frac{\pi }{4}\text{, }\pm \frac{3\pi }{4}\text{ } \\ & \text{Therefore} \\ & \text{We have the critical points} \\ & x=\pm \frac{\pi }{4}\text{, }\pm \frac{3\pi }{4}\text{ } \\ & \text{Evaluating }f\text{ at each of these points, we have} \\ & f\left( -\pi \right)=\sin 2\left( -\pi \right)+3=3 \\ & f\left( -\frac{3\pi }{4} \right)=\sin 2\left( -\frac{3\pi }{4} \right)+3=4 \\ & f\left( -\frac{\pi }{4} \right)=\sin 2\left( -\frac{\pi }{4} \right)+3=2 \\ & f\left( \frac{\pi }{4} \right)=\sin 2\left( \frac{\pi }{4} \right)+3=4 \\ & f\left( -\frac{3\pi }{4} \right)=\sin 2\left( -\frac{3\pi }{4} \right)+3=2 \\ & f\left( \pi \right)=\sin 2\left( \pi \right)+3=3 \\ & \text{The largest of those function values are:} \\ & f\left( -\frac{3\pi }{4} \right)=4\text{ and }f\left( \frac{\pi }{4} \right)=4 \\ & \text{Which are absolute }\left( \text{and local} \right)\text{ maximum on }\left[ -\pi ,\pi \right] \\ & \text{The smallest of those function values are:} \\ & f\left( -\frac{\pi }{4} \right)=2\text{ and }f\left( \frac{3\pi }{4} \right)=2 \\ & \text{Which are absolute }\left( \text{and local} \right)\text{ minimum on }\left[ -\pi ,\pi \right] \\ & \\ & \text{Absolute maximum: }\left( -\frac{3\pi }{4},4 \right)\text{ and }\left( \frac{\pi }{4},4 \right) \\ & \text{Absolute minimum: }\left( -\frac{\pi }{4},2 \right)\text{ and }\left( \frac{3\pi }{4},2 \right) \\ \end{align}\]