1. **Problem Statement:** We analyze the function $f$ defined on the interval $[1,5]$ and determine which of the given statements (a) to (d) about its critical points, inflection points, absolute maximum, and convexity are correct or not. 2. **Definitions and Formulas:** - A **critical point** occurs where the first derivative $f'(x) = 0$ or is undefined. - An **inflection point** occurs where the second derivative $f''(x) = 0$ and the concavity changes. - An **absolute maximum** is the highest value of $f(x)$ on the interval. - The function is **convex upward** (concave up) where $f''(x) > 0$. 3. **Given Information from the Graph:** - Critical points approximately at $x=1.5$, $x=3$, and $x=4.5$ (3 points). - Inflection points approximately at $x=2$ and $x=4$ (2 points). - The highest peak near $x=4$ suggests an absolute maximum. - The curve changes concavity at inflection points, so convexity varies. 4. **Check Each Statement:** (a) "The function $f$ has three critical points." - From the graph, there are 3 critical points at $x=1.5$, $3$, and $4.5$. - So, (a) is **correct**. (b) "The function $f$ has two inflection points." - The graph shows two inflection points at $x=2$ and $x=4$. - So, (b) is **correct**. (c) "The function $f$ has an absolute maximum value." - The highest peak near $x=4$ is the absolute maximum on $[1,5]$. - So, (c) is **correct**. (d) "The curve of the function $f$ is convex upward on the interval $]1,5[$." - The function changes concavity at inflection points, so it is not convex upward on the entire interval $]1,5[$. - It is convex upward only on subintervals where $f''(x) > 0$, not the whole interval. - So, (d) is **not correct**. **Final answer:** The statement that is not correct is (d).