1. We are given the function $f(x) = x^2 - 3x + 8$ and need to find: (a) intervals where $f$ is increasing, (b) intervals where $f$ is decreasing, (c) intervals where $f$ is concave up, (d) intervals where $f$ is concave down, (e) $x$-coordinates of inflection points. 2. To analyze increasing/decreasing behavior, use the first derivative test: $$f'(x) = \frac{d}{dx}(x^2 - 3x + 8) = 2x - 3$$ 3. To analyze concavity and inflection points, use the second derivative: $$f''(x) = \frac{d}{dx}(2x - 3) = 2$$ 4. Find critical points by setting $f'(x) = 0$: $$2x - 3 = 0 \implies x = \frac{3}{2}$$ 5. Determine intervals of increase/decrease: - For $x < \frac{3}{2}$, $f'(x) = 2x - 3 < 0$ so $f$ is decreasing. - For $x > \frac{3}{2}$, $f'(x) = 2x - 3 > 0$ so $f$ is increasing. 6. Since $f''(x) = 2 > 0$ for all $x$, $f$ is concave up everywhere. 7. Because $f''(x)$ never changes sign, there are no inflection points. Final answers: (a) Increasing on $(\frac{3}{2}, \infty)$ (b) Decreasing on $(-\infty, \frac{3}{2})$ (c) Concave up on $(-\infty, \infty)$ (d) Concave down on $\varnothing$ (no intervals) (e) No inflection points