Chapter 3.3, Problem 41E

(a)

To determine

The open interval in which the function,$f\left(x\right)=\frac{x}{2}+\cos x$, $0<x<2\pi$, is increasing or decreasing.

(b)

To determine

The relative extrema of the function, $f\left(x\right)=\frac{x}{2}+\cos x$, $0<x<2\pi$, using first derivative test.

(c)

To determine

To prove: The function, $f\left(x\right)=\frac{x}{2}+\cos x$, $0<x<2\pi$, is increasing on the intervals $\left(0,\frac{\pi }{6}\right)$ and $\left(\frac{5\pi }{6},2\pi \right)$, and decreasing on the interval $\left(\frac{\pi }{6},\frac{5\pi }{6}\right)$, and has relative maximum at $x=\frac{\pi }{6}$ and relative minimum at $x=\frac{5\pi }{6}$, using graphing utility.