Chapter 4 - Applications of the Derivative - 4.4 The Shape of a Graph - Exercises - Page 194: 8

concave up $\frac{\pi}{4}$ < $θ$ < $0$, $\frac{3\pi}{4}$ < $θ$ < $\pi$ concave down $\frac{\pi}{4}$ < $θ$ < $\frac{3\pi}{4}$ inflection point at $θ = \frac{\pi}{4}, \frac{3\pi}{4}$

Let $f(θ)$ = $θ+sin^{2}θ$ $f'(θ)$ = $1+2sinθcosθ$ = $1+sin2θ$ $f''(θ)$ = $2cos2θ$ concave up $2cos2θ$ > $0$ $cos2θ$ > $0$ $\frac{\pi}{2}$ < $2θ$ < $0$, $\frac{3\pi}{2}$ < $2θ$ < $2\pi$ $\frac{\pi}{4}$ < $θ$ < $0$, $\frac{3\pi}{4}$ < $θ$ < $\pi$ concave down $\frac{\pi}{4}$ < $θ$ < $\frac{3\pi}{4}$ inflection point at $θ = \frac{\pi}{4}, \frac{3\pi}{4}$