1. **Problem statement:** Given the function $y = \sin(2x + 1)$ for $0 \leq x \leq \pi$, find the $x$-coordinates of the maximum and minimum points of $y$. 2. **Formula and rules:** The sine function $\sin(\theta)$ has maximum value 1 and minimum value -1. To find maxima and minima of $y = \sin(2x + 1)$, solve for $2x + 1 = \frac{\pi}{2} + 2k\pi$ (maxima) and $2x + 1 = \frac{3\pi}{2} + 2k\pi$ (minima), where $k$ is an integer. 3. **Find maxima:** $$2x + 1 = \frac{\pi}{2}$$ $$2x = \frac{\pi}{2} - 1$$ $$x = \frac{\pi}{4} - \frac{1}{2}$$ Calculate numerically: $$x \approx \frac{3.1416}{4} - 0.5 = 0.7854 - 0.5 = 0.2854$$ 4. **Find minima:** $$2x + 1 = \frac{3\pi}{2}$$ $$2x = \frac{3\pi}{2} - 1$$ $$x = \frac{3\pi}{4} - \frac{1}{2}$$ Calculate numerically: $$x \approx \frac{3 \times 3.1416}{4} - 0.5 = 2.3562 - 0.5 = 1.8562$$ 5. **Check domain:** Both $x \approx 0.3$ and $x \approx 1.9$ lie within $0 \leq x \leq \pi$ (since $\pi \approx 3.1416$). 6. **Final answers:** - Maximum at $x = 0.3$ (to 1 decimal place) - Minimum at $x = 1.9$ (to 1 decimal place)