1. **Problem Statement:** We are given that for each $x \in [a,b]$, the first derivative $f'(x) < 0$ and the second derivative $f''(x) > 0$. We need to determine which curve among the given options represents the function $f$ on the interval $[a,b]$. 2. **Understanding the derivatives:** - $f'(x) < 0$ means the function $f$ is **decreasing** on $[a,b]$. - $f''(x) > 0$ means the function $f$ is **concave up** (convex) on $[a,b]$. 3. **Interpreting the conditions:** - Since $f'(x) < 0$, the function must be going down as $x$ moves from $a$ to $b$. - Since $f''(x) > 0$, the slope $f'(x)$ is increasing (becoming less negative) as $x$ increases, so the curve is bending upwards. 4. **Analyzing the curves:** - Curve (a): Starts high and decreases, then slightly decreases further. This matches decreasing behavior. - Curve (b): Starts low and increases sharply, so $f'(x) > 0$, which contradicts $f'(x) < 0$. - Curve (c): Starts high and decreases below zero, which matches decreasing behavior. - Curve (d): Starts near zero and increases steeply, so $f'(x) > 0$, contradicting $f'(x) < 0$. 5. **Checking concavity for curves (a) and (c):** - Curve (a) is decreasing and appears to be concave up (bending upwards), consistent with $f''(x) > 0$. - Curve (c) is decreasing but appears concave down (bending downwards), inconsistent with $f''(x) > 0$. 6. **Conclusion:** The curve that is decreasing and concave up is curve (a). **Final answer:** Curve (a) represents the function $f$ on $[a,b]$. $$\boxed{\text{Curve (a)}}$$