# /// 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：配方法求顶点
# y = -2(x² - 4x) - 3
# y = -2(x² - 4x + 4 - 4) - 3
# y = -2(x - 2)² + 8 - 3
# y = -2(x - 2)² + 5

# 方法2：使用公式 x_vertex = -b/(2a)
a = -2
b = 8
c = -3

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

print(f"\n1. 顶点坐标")
print(f"   使用公式 x = -b/(2a) = -{b}/(2×{a}) = {x_vertex}")
print(f"   代入求 y = {a}×{x_vertex}² + {b}×{x_vertex} + ({c}) = {y_vertex}")
print(f"   顶点坐标: ({int(x_vertex)}, {int(y_vertex)})")

print(f"\n2. 函数最大值")
print(f"   由于 a = {a} < 0，抛物线开口向下")
print(f"   函数在顶点处取得最大值")
print(f"   最大值: y_max = {int(y_vertex)}")

# 使用 SymPy 验证
print("\n" + "=" * 50)
print("SymPy 符号计算验证")
print("=" * 50)

# 求导数找极值点
dy = sp.diff(y, x)
critical_points = sp.solve(dy, x)
print(f"\n导数: y' = {dy}")
print(f"令 y' = 0，解得 x = {critical_points}")

# 验证是最大值（二阶导数 < 0）
d2y = sp.diff(dy, x)
print(f"二阶导数: y'' = {d2y} < 0，确认为最大值点")

# 计算顶点处的函数值
x_v = critical_points[0]
y_v = y.subs(x, x_v)
print(f"\n顶点坐标: ({x_v}, {y_v})")
print(f"最大值: {y_v}")

# 配方形式
print("\n" + "=" * 50)
print("配方形式验证")
print("=" * 50)
vertex_form = -2*(x - 2)**2 + 5
expanded = sp.expand(vertex_form)
print(f"配方形式: y = -2(x - 2)² + 5")
print(f"展开验证: {expanded}")
print(f"与原式相等: {sp.simplify(expanded - y) == 0}")