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