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{ "quality": "incomplete", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Curve Sketching" ], "new_concepts": [ "Function Parity" ], "current_concepts": [ "Curve Sketching", "Function Parity" ] } ## 分析函数的奇偶性 Using the *Curve Sketching* knowledge point $$ \begin{aligned} &f(x) = x \sin x \ &f(-x) = (-x) \sin(-x) = (-x)(-\sin x) = x \sin x = f(x) \end{aligned} $$ 因此，该函数为偶函数，其图象关于 $y$ 轴对称。由此可排除 B 和 D。 ## 分析函数在原点附近的符号 Using the *Curve Sketching* knowledge point $$ \begin{aligned} &\text{当 } x \in (0, \pi) \text{ 时，} x > 0 \text{ 且 } \sin x > 0 \ &f(x) = x \sin x > 0 \end{aligned} $$ 因此，在原点两侧的邻域内，函数值应大于零，图象位于 $x$ 轴上方。由此可排除 C。 ## 确定最终选项综合上述分析，函数图象关于 $y$ 轴对称，且在原点附近位于 $x$ 轴上方，符合此特征的图象为 A。 ## Question 1 (A) 关于 $y$ 轴对称且在原点附近图象在 $x$ 轴上方

(B) 非对称图象

(C) 关于 $y$ 轴对称但在原点附近图象在 $x$ 轴下方

(D) 关于原点对称的奇函数图象

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Explanation:

分析函数的奇偶性

分析函数在原点附近的符号

确定最终选项

Answer:

Question 1