01-基石题-test

01-基石题-test/模型qwen3-coder-plus工具iflow/solve.py

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+# /// script
+# requires-python = ">=3.11"
+# dependencies = ["sympy", "numpy", "matplotlib"]
+# ///
+
+import sympy as sp
+import numpy as np
+
+# 定义变量
+x = sp.symbols('x', real=True)
+
+# 定义二次函数 y = -2x^2 + 8x - 3
+func = -2*x**2 + 8*x - 3
+
+print("给定二次函数: y =", func)
+
+# 方法1: 使用导数求顶点
+# 顶点的x坐标是导数为0的点
+derivative = sp.diff(func, x)
+print("函数的一阶导数:", derivative)
+
+# 求导数为0的点
+critical_points = sp.solve(derivative, x)
+print("导数为0的点:", critical_points)
+
+# 顶点的x坐标
+vertex_x = critical_points[0]
+vertex_y = func.subs(x, vertex_x)
+
+print(f"顶点坐标: ({vertex_x}, {vertex_y})")
+
+# 判断最大值还是最小值
+second_derivative = sp.diff(derivative, x)
+print("函数的二阶导数:", second_derivative)
+
+if second_derivative < 0:
+    print("由于二次项系数为负数，函数开口向下，顶点为最大值点")
+    print(f"函数的最大值: {vertex_y}")
+elif second_derivative > 0:
+    print("由于二次项系数为正数，函数开口向上，顶点为最小值点")
+    print(f"函数的最小值: {vertex_y}")
+else:
+    print("这不是二次函数")
+
+# 方法2: 使用二次函数顶点公式验证
+# 对于二次函数 y = ax^2 + bx + c，顶点x坐标为 x = -b/(2a)
+a = -2
+b = 8
+c = -3
+
+vertex_x_formula = -b / (2*a)
+vertex_y_formula = a * vertex_x_formula**2 + b * vertex_x_formula + c
+
+print("\n使用顶点公式验证:")
+print(f"顶点x坐标: x = -b/(2a) = {-b}/(2*{a}) = {vertex_x_formula}")
+print(f"顶点y坐标: y = {a}*({vertex_x_formula})^2 + {b}*({vertex_x_formula}) + {c} = {vertex_y_formula}")
+
+# 最终结果
+print("\n" + "="*40)
+print("最终答案:")
+print(f"1. 函数的顶点坐标: ({vertex_x}, {vertex_y})")
+print(f"2. 函数的最大值: {vertex_y}")
+print("="*40)