# /// script
# requires-python = ">=3.11"
# dependencies = ["sympy"]
# ///

import sympy as sp

# 定义符号（实数）
x = sp.symbols("x", real=True)

# 定义函数
y = -2 * x**2 + 8 * x - 3

# 方法1：通过顶点公式求解
# 二次函数 y = ax^2 + bx + c 的顶点公式：x = -b/(2a)
a = -2
b = 8
c = -3

x_vertex = -b / (2 * a)
y_vertex = a * x_vertex**2 + b * x_vertex + c

# 方法2：通过求导验证
dy_dx = sp.diff(y, x)
critical_points = sp.solve(dy_dx, x)
d2y_dx2 = sp.diff(dy_dx, x)

print("=" * 50)
print("二次函数：y = -2x² + 8x - 3")
print("=" * 50)

print("\n【方法1：顶点公式】")
print(f"顶点坐标：({x_vertex}, {y_vertex})")

print("\n【方法2：求导法】")
print(f"导数：y' = {dy_dx}")
print(f"二阶导数：y'' = {d2y_dx2}")
print(f"临界点：x = {critical_points[0]}")
print(f"代入原函数：y = {y.subs(x, critical_points[0])}")
print(f"二阶导数：{d2y_dx2} < 0，故为极大值点")

print("\n【结论】")
print(f"1. 顶点坐标：({x_vertex}, {y_vertex})")
print(f"2. 函数最大值：{y_vertex}")