1. **State the problem:** Find critical points, concavity intervals, points of inflection, and classify critical points for $$f(x) = 3x + 3\sin(x)\text{ on }[0,2\pi].$$ 2. **Find the first derivative:** $$f'(x) = \frac{d}{dx}(3x) + \frac{d}{dx}(3\sin(x)) = 3 + 3\cos(x).$$ 3. **Find critical points:** Solve $$f'(x) = 0$$: $$3 + 3\cos(x) = 0 \implies \cos(x) = -1.$$ Over $$[0, 2\pi]$$, $$\cos(x) = -1$$ at $$x = \pi.$$ So the only critical point is at $$x=\pi$$. 4. **Find the second derivative:** $$f''(x) = \frac{d}{dx}f'(x) = \frac{d}{dx}(3 + 3\cos(x)) = -3\sin(x).$$ 5. **Evaluate second derivative at critical point:** $$f''(\pi) = -3\sin(\pi) = 0.$$ Since $$f''(\pi) = 0$$, the Second Derivative Test is inconclusive here. 6. **Analyze the first derivative sign to classify critical point:** - For $$x < \pi$$, pick $$x = \frac{\pi}{2}$$: $$f'(\frac{\pi}{2}) = 3 + 3\cos(\frac{\pi}{2}) = 3 + 0 =3>0,$$ so $$f' > 0$$ to the left. - For $$x > \pi$$, pick $$x = \frac{3\pi}{2}$$: $$f'(\frac{3\pi}{2}) = 3 + 3\cos(\frac{3\pi}{2}) = 3 + 0=3>0,$$ so $$f' > 0$$ to the right. The function is increasing on both sides of $$x=\pi$$, so $$x=\pi$$ is neither a local max nor min. 7. **Find points of inflection:** Solve $$f''(x) = 0$$: $$-3\sin(x) = 0 \implies \sin(x) = 0.$$ On $$[0, 2\pi]$$, $$\sin(x) = 0$$ at $$x=0, \pi, 2\pi.$$ 8. **Determine concavity intervals:** - For $$x \in (0, \pi)$$, pick $$x = \frac{\pi}{2}$$: $$f''(\frac{\pi}{2}) = -3\sin(\frac{\pi}{2}) = -3(1) = -3 < 0,$$ so $$f$$ is concave down on $$(0, \pi).$$ - For $$x \in (\pi, 2\pi)$$, pick $$x = \frac{3\pi}{2}$$: $$f''(\frac{3\pi}{2}) = -3\sin(\frac{3\pi}{2}) = -3(-1) = 3 > 0,$$ so $$f$$ is concave up on $$(\pi, 2\pi).$$ 9. **Summary:** - Critical point: $$\pi, f(\pi) = 3\pi + 3\sin(\pi) = 3\pi + 0 = 3\pi.$$ - Second Derivative Test inconclusive at $$\pi$$. - Increasing on both sides of $$\pi$$, so neither local max nor min. - Inflection points: $$x = 0, \pi, 2\pi.$$ - Concave down on $$(0, \pi)$$. - Concave up on $$(\pi, 2\pi)$$. 10. **Intervals and behaviors around critical and inflection points:** - On $$(0, \pi)$$ to the left of critical point $$\pi$$, $$f$$ is concave down and $$f' > 0$$ (increasing). - On $$(\pi, 2\pi)$$ to the right of the critical point $$\pi$$, $$f$$ is concave up and $$f' > 0$$ (increasing). - On $$(0, \pi)$$ to the left of inflection point $$\pi$$, $$f$$ is concave down. - On $$(\pi, 2\pi)$$ to the right of inflection point $$\pi$$, $$f$$ is concave up.