01-基石题-test

01-基石题-test/模型gemini-claude-opus-4-5-thinking工具GithubCopilot/solve.py

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+# /// script
+# requires-python = ">=3.11"
+# dependencies = ["sympy"]
+# ///
+
+import sympy as sp
+
+# 定义符号变量
+x = sp.symbols('x', real=True)
+
+# 定义二次函数
+y = -2*x**2 + 8*x - 3
+
+print("=" * 50)
+print("二次函数 y = -2x² + 8x - 3 求解")
+print("=" * 50)
+
+# 方法1：配方法求顶点
+# y = -2(x² - 4x) - 3
+# y = -2(x² - 4x + 4 - 4) - 3
+# y = -2(x - 2)² + 8 - 3
+# y = -2(x - 2)² + 5
+
+# 方法2：使用公式 x_vertex = -b/(2a)
+a = -2
+b = 8
+c = -3
+
+x_vertex = -b / (2*a)
+y_vertex = a * x_vertex**2 + b * x_vertex + c
+
+print(f"\n1. 顶点坐标")
+print(f"   使用公式 x = -b/(2a) = -{b}/(2×{a}) = {x_vertex}")
+print(f"   代入求 y = {a}×{x_vertex}² + {b}×{x_vertex} + ({c}) = {y_vertex}")
+print(f"   顶点坐标: ({int(x_vertex)}, {int(y_vertex)})")
+
+print(f"\n2. 函数最大值")
+print(f"   由于 a = {a} < 0，抛物线开口向下")
+print(f"   函数在顶点处取得最大值")
+print(f"   最大值: y_max = {int(y_vertex)}")
+
+# 使用 SymPy 验证
+print("\n" + "=" * 50)
+print("SymPy 符号计算验证")
+print("=" * 50)
+
+# 求导数找极值点
+dy = sp.diff(y, x)
+critical_points = sp.solve(dy, x)
+print(f"\n导数: y' = {dy}")
+print(f"令 y' = 0，解得 x = {critical_points}")
+
+# 验证是最大值（二阶导数 < 0）
+d2y = sp.diff(dy, x)
+print(f"二阶导数: y'' = {d2y} < 0，确认为最大值点")
+
+# 计算顶点处的函数值
+x_v = critical_points[0]
+y_v = y.subs(x, x_v)
+print(f"\n顶点坐标: ({x_v}, {y_v})")
+print(f"最大值: {y_v}")
+
+# 配方形式
+print("\n" + "=" * 50)
+print("配方形式验证")
+print("=" * 50)
+vertex_form = -2*(x - 2)**2 + 5
+expanded = sp.expand(vertex_form)
+print(f"配方形式: y = -2(x - 2)² + 5")
+print(f"展开验证: {expanded}")
+print(f"与原式相等: {sp.simplify(expanded - y) == 0}")