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