45 lines
1.1 KiB
Python
45 lines
1.1 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（实数）
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x = sp.symbols("x", real=True)
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# 定义函数
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y = -2 * x**2 + 8 * x - 3
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# 方法1：通过顶点公式求解
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# 二次函数 y = ax^2 + bx + c 的顶点公式：x = -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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# 方法2：通过求导验证
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dy_dx = sp.diff(y, x)
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critical_points = sp.solve(dy_dx, x)
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d2y_dx2 = sp.diff(dy_dx, x)
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print("=" * 50)
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print("二次函数：y = -2x² + 8x - 3")
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print("=" * 50)
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print("\n【方法1：顶点公式】")
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print(f"顶点坐标：({x_vertex}, {y_vertex})")
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print("\n【方法2：求导法】")
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print(f"导数：y' = {dy_dx}")
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print(f"二阶导数：y'' = {d2y_dx2}")
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print(f"临界点：x = {critical_points[0]}")
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print(f"代入原函数：y = {y.subs(x, critical_points[0])}")
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print(f"二阶导数：{d2y_dx2} < 0，故为极大值点")
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print("\n【结论】")
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print(f"1. 顶点坐标：({x_vertex}, {y_vertex})")
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print(f"2. 函数最大值：{y_vertex}")
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