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

import sympy as sp

# 定义符号变量
x = sp.symbols('x', real=True)

# 定义二次函数
y = -2*x**2 + 8*x - 3

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

# 方法1：使用配方法/顶点公式
# 对于 y = ax² + bx + c，顶点为 (-b/(2a), (4ac-b²)/(4a))
a, b, c = -2, 8, -3

# 顶点横坐标
x_vertex = -b / (2*a)
# 顶点纵坐标（最值）
y_vertex = (4*a*c - b**2) / (4*a)

print(f"\n【方法1：顶点公式】")
print(f"a = {a}, b = {b}, c = {c}")
print(f"顶点横坐标 x = -b/(2a) = -{b}/(2×{a}) = {x_vertex}")
print(f"顶点纵坐标 y = (4ac-b²)/(4a) = {y_vertex}")

# 方法2：使用 SymPy 求导
print(f"\n【方法2：SymPy 求导验证】")
dy = sp.diff(y, x)
print(f"y' = {dy}")

# 令导数为0，求驻点
critical_points = sp.solve(dy, x)
print(f"令 y' = 0，解得 x = {critical_points}")

if critical_points:
    x_val = critical_points[0]
    y_val = y.subs(x, x_val)
    print(f"将 x = {x_val} 代入原函数：y = {y_val}")

# 方法3：配方法展示
print(f"\n【方法3：配方法】")
print("y = -2x² + 8x - 3")
print("  = -2(x² - 4x) - 3")
print("  = -2(x² - 4x + 4 - 4) - 3")
print("  = -2(x - 2)² + 8 - 3")
print("  = -2(x - 2)² + 5")
print("顶点形式：y = -2(x - 2)² + 5")

# 最终结果
print("\n" + "=" * 50)
print("【最终结果】")
print("=" * 50)
print(f"顶点坐标：({int(x_vertex)}, {int(y_vertex)})")
print(f"由于 a = {a} < 0，抛物线开口向下")
print(f"函数最大值：y_max = {int(y_vertex)}（在 x = {int(x_vertex)} 处取得）")