1. **Problem statement:** We are given the graph of the first derivative $f'$ of a continuous function $f$ on $\mathbb{R}$. We need to identify which statement among (a), (b), (c), and (d) is wrong. 2. **Recall key concepts:** - A critical point of $f$ occurs where $f'(x) = 0$ or $f'(x)$ is undefined. - An inflection point of $f$ occurs where the concavity changes, i.e., where $f''(x)$ changes sign. - The convexity (concavity) of $f$ is determined by the sign of $f''(x)$: if $f''(x) > 0$, $f$ is convex up; if $f''(x) < 0$, $f$ is convex down. 3. **Analyze the graph of $f'$:** - The graph of $f'$ is symmetric around the y-axis and sharply rises near $x=0$. - At $x=0$, $f'(0) = 0$ because the curve touches the x-axis. 4. **Check each statement:** (a) "The function $f$ has an inflection point at $x=0$": - Inflection points occur where $f''$ changes sign. - Since $f'$ has a minimum at $x=0$ (sharp rise on both sides), $f''(0) = 0$ and $f''$ changes from negative to positive or vice versa. - So, (a) is true. (b) "The curve of $f$ is convex up on $]-\infty,0[$": - Convex up means $f''(x) > 0$. - Since $f'$ is decreasing on $]-\infty,0[$ (curve goes down to 0), $f''(x) = f'(x)' < 0$ there. - So $f$ is convex down on $]-\infty,0[$, not convex up. - Therefore, (b) is false. (c) "The function $f$ has a critical point at $x=0$": - Since $f'(0) = 0$, $x=0$ is a critical point. - So, (c) is true. (d) "The curve of $f$ is convex up on $]0,\infty[$": - For $x>0$, $f'$ is increasing (curve rises sharply), so $f''(x) > 0$. - Hence, $f$ is convex up on $]0,\infty[$. - So, (d) is true. 5. **Conclusion:** The wrong statement is (b). **Final answer:** (b) The curve of the function $f$ is convex up on the interval $]-\infty,0[$ is wrong.