# /// 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)