1. **State the problem:** Given a twice-differentiable function $f(x)$ with $f''(x) = 0$ for all $x$ in the domain, determine which statements must be true. 2. **Recall the meaning of the second derivative:** The second derivative $f''(x)$ measures the concavity of $f(x)$. If $f''(x) = 0$ everywhere, the function has zero concavity and is linear. 3. **Use the fact that $f''(x) = 0$:** Integrate once: $$f''(x) = 0 \implies f'(x) = C_1,$$ where $C_1$ is a constant. 4. **Integrate again:** $$f'(x) = C_1 \implies f(x) = C_1 x + C_2,$$ where $C_2$ is another constant. 5. **Interpretation:** $f(x)$ is a polynomial of degree 1 (a linear function) on the entire domain. 6. **Check the options:** - $f(x)$ is a degree 1 polynomial on the entire domain: **True**. - $f(x)$ has no critical points: Critical points occur where $f'(x) = 0$. Since $f'(x) = C_1$, if $C_1 \neq 0$, no critical points; if $C_1 = 0$, $f(x)$ is constant and every point is critical. So this is **not necessarily true**. - $f(x)$ has an inflection point at $x=0$: Inflection points require a change in concavity, but $f''(x) = 0$ everywhere, so **no inflection points**. - $f(x)$ is constant on the entire domain: Only if $C_1 = 0$, so **not necessarily true**. **Final answer:** The only statement that must be true is that $f(x)$ is a degree 1 polynomial on the entire domain.