1. **Problem statement:** We analyze the function $f$ defined on the interval $[1,5]$ based on the given graph and determine which statement among (a), (b), (c), and (d) is not correct. 2. **Understanding critical points:** Critical points occur where the derivative $f'(x)$ is zero or undefined, typically at local maxima, minima, or horizontal inflection points. The graph shows three such points where the slope changes sign, so (a) is correct. 3. **Inflection points:** Inflection points occur where the concavity changes, i.e., where the second derivative $f''(x)$ changes sign. The graph suggests two such points where the curve changes from concave up to concave down or vice versa, so (b) is correct. 4. **Absolute maximum:** The graph shows a local maximum near $x=4$, but since the function is defined on a closed interval $[1,5]$, the absolute maximum must be the highest value on this interval. The graph touches the x-axis at $x=1$ and $x=5$ and the local maximum near 4 is higher than these endpoints, so (c) is correct. 5. **Convexity on the interval $(1,5)$:** The statement (d) claims the curve is convex upward on the entire open interval $(1,5)$. However, the graph shows regions where the curve is concave downward (convex downward), so (d) is not correct. **Final answer:** The statement that is not correct is (d). $$\boxed{\text{The incorrect statement is (d).}}$$