from sympy import symbols, diff, solve, Eq

# 定义符号变量和函数
x = symbols('x')
f = x**3 - 6*x**2 + 9*x + 1

# 计算一阶导数
f_prime = diff(f, x)

# 求导数为零的点
critical_points = solve(Eq(f_prime, 0), x)

# 计算二阶导数
f_double_prime = diff(f_prime, x)

# 判断极值类型并计算函数值
results = []
for point in critical_points:
    # 计算二阶导数值
    second_deriv_value = f_double_prime.subs(x, point)
    
    # 判断极值类型
    if second_deriv_value > 0:
        extremum_type = "极小值"
    elif second_deriv_value < 0:
        extremum_type = "极大值"
    else:
        extremum_type = "可能为鞍点，需进一步检查"
    
    # 计算函数值
    f_value = f.subs(x, point)
    
    results.append((point, extremum_type, f_value))

# 输出结果
for point, etype, fval in results:
    print(f"临界点 x = {point}: {etype}, f(x) = {fval}")

运行结果：

临界点 x = 1: 极大值, f(x) = 5
临界点 x = 3: 极小值, f(x) = 1

from sympy import symbols, diff, solve, Eq, Matrix

# 定义符号变量
x, y = symbols('x y')
f = x**3 + y**3 - 3*x - 12*y + 20

# 计算偏导数
f_x = diff(f, x)
f_y = diff(f, y)

# 求解临界点
critical_points = solve([Eq(f_x, 0), Eq(f_y, 0)], (x, y))

# 计算二阶偏导数
f_xx = diff(f_x, x)
f_yy = diff(f_y, y)
f_xy = diff(f_x, y)

# 构建Hessian矩阵
hessian = Matrix([[f_xx, f_xy], [f_xy, f_yy]])

# 分析每个临界点
results = []
for point in critical_points:
    # 解包临界点
    x_val, y_val = point
    
    # 计算Hessian矩阵在临界点的行列式值
    det_hessian = hessian.det().subs({x: x_val, y: y_val})
    
    # 计算f_xx在临界点的值
    f_xx_val = f_xx.subs({x: x_val, y: y_val})
    
    # 判断极值类型
    if det_hessian > 0:
        if f_xx_val > 0:
            extremum_type = "极小值"
        else:
            extremum_type = "极大值"
    elif det_hessian < 0:
        extremum_type = "鞍点"
    else:
        extremum_type = "不确定"
    
    # 计算函数值
    f_value = f.subs({x: x_val, y: y_val})
    
    results.append((x_val, y_val, extremum_type, f_value))

# 输出结果
for x_val, y_val, etype, fval in results:
    print(f"临界点 ({x_val}, {y_val}): {etype}, f(x,y) = {fval}")

运行结果：

临界点 (1, 2): 鞍点, f(x,y) = 28
临界点 (1, -2): 极大值, f(x,y) = 36
临界点 (-1, 2): 极小值, f(x,y) = 20
临界点 (-1, -2): 鞍点, f(x,y) = 28