1. **Problem Statement:** Given that $f'(x) < 0$ and $f''(x) > 0$ for each $x \in [a,b]$, determine which graph represents the function $f$ on the interval $[a,b]$. 2. **Understanding the conditions:** - $f'(x) < 0$ means the function $f$ is **decreasing** on $[a,b]$. - $f''(x) > 0$ means the function $f$ is **concave upward** (convex) on $[a,b]$. 3. **Interpreting the graphs:** - Graph (a): Decreasing and concave upward. - Graph (b): Increasing and concave upward. - Graph (c): Decreasing and concave downward. - Graph (d): Increasing and concave upward. 4. **Matching conditions:** Since $f'(x) < 0$ (decreasing) and $f''(x) > 0$ (concave upward), the correct graph must be decreasing and concave upward. 5. **Conclusion:** Graph (a) matches these conditions. **Final answer:** The curve of the function $f$ on $[a,b]$ is represented by **Graph (a)**.