求x(x+1)(x+2)(x+3)(x+4)(x+5)的最小值

ydwb

求x(x+1)(x+2)(x+3)(x+4)(x+5)的最小值

爱吃榴莲

不会

空python

最小值可能是负数

ydwb

@FishC，请回答问题：求x(x+1)(x+2)(x+3)(x+4)(x+5)的最小值

ydwb

1. import numpy as np
   
2. import matplotlib.pyplot as plt
   
3. 
   
4. # ============ 0. 中文字体配置 ============
   
5. plt.rcParams['font.sans-serif'] = ['SimHei', 'Microsoft YaHei', 'Arial Unicode MS']
   
6. plt.rcParams['axes.unicode_minus'] = False
   
7. 
   
8. # ============ 1. 定义原函数 f(x) ============
   
9. def f(x):
   
10.     return x * (x+1) * (x+2) * (x+3) * (x+4) * (x+5)
    
11. 
    
12. # ============ 2. 对称换元：t = x + 2.5, u = t^2 ============
    
13. def g(u):
    
14.     # g(u) = (u - 6.25)(u - 2.25)(u - 0.25)
    
15.     return (u - 6.25) * (u - 2.25) * (u - 0.25)
    
16. 
    
17. # ============ 3. 精确解析解 ============
    
18. # 对 g(u) 求导：g'(u) = 3u^2 - 17.5u + 16.1875 = 0
    
19. # 极小值点 u = (35 + 8*sqrt(7)) / 12
    
20. u_min_exact = (35 + 8*np.sqrt(7)) / 12
    
21. g_min_exact = (u_min_exact - 6.25) * (u_min_exact - 2.25) * (u_min_exact - 0.25)
    
22. 
    
23. t_min = np.sqrt(u_min_exact)
    
24. x1_exact = -2.5 - t_min
    
25. x2_exact = -2.5 + t_min
    
26. 
    
27. print("=" * 55)
    
28. print("【解析解】")
    
29. print("=" * 55)
    
30. print(f"极小值点 u* = (35 + 8√7)/12 ≈ {u_min_exact:.6f}")
    
31. print(f"最小值 g(u*) = (-160 - 112√7)/27 ≈ {g_min_exact:.6f}")
    
32. print(f"对应 t = ±√u* ≈ ±{t_min:.6f}")
    
33. print(f"对应 x1 = -2.5 - √u* ≈ {x1_exact:.6f}")
    
34. print(f"对应 x2 = -2.5 + √u* ≈ {x2_exact:.6f}")
    
35. 
    
36. # ============ 4. 数值验证 ============
    
37. print("\n" + "=" * 55)
    
38. print("【数值验证】")
    
39. print("=" * 55)
    
40. print(f"f(x1) = {f(x1_exact):.8f}")
    
41. print(f"f(x2) = {f(x2_exact):.8f}")
    
42. print(f"f(-2.5) = {f(-2.5):.8f}    (对称中心，非最小值)")
    
43. print(f"f(-4.66) = {f(-4.66):.8f}  (近似点)")
    
44. print(f"f(-4.67) = {f(-4.67):.8f}")
    
45. print(f"f(-4.65) = {f(-4.65):.8f}")
    
46. 
    
47. # 用 scipy 数值求最小值做交叉验证
    
48. try:
    
49.     from scipy.optimize import minimize_scalar
    
50.     res = minimize_scalar(f, bounds=(-5, 0), method='bounded')
    
51.     print(f"\nscipy 数值最小值: x ≈ {res.x:.6f}, f(x) ≈ {res.fun:.8f}")
    
52. except ImportError:
    
53.     print("\n(未安装 scipy，跳过数值优化验证)")
    
54. 
    
55. # ============ 5. 图形 ============
    
56. fig, axes = plt.subplots(1, 2, figsize=(14, 5.5))
    
57. 
    
58. # ---- 图1：原函数 f(x) ----
    
59. x = np.linspace(-6, 1, 2000)
    
60. y = f(x)
    
61. 
    
62. ax1 = axes[0]
    
63. ax1.plot(x, y, color='blue', linewidth=2, label=r'$f(x)=x(x+1)(x+2)(x+3)(x+4)(x+5)$')
    
64. ax1.axhline(0, color='black', linewidth=0.8, linestyle='--')
    
65. ax1.axvline(-2.5, color='gray', linewidth=0.8, linestyle=':', label='x = -2.5 (对称中心)')
    
66. 
    
67. # 标出整数根
    
68. for r in [-5, -4, -3, -2, -1, 0]:
    
69.     ax1.plot(r, 0, 'ro', markersize=5)
    
70.     ax1.text(r, 3, f'{r}', ha='center', color='red', fontsize=9)
    
71. 
    
72. # 标出最小值点
    
73. ax1.plot(x1_exact, g_min_exact, 'mo', markersize=10, zorder=5)
    
74. ax1.annotate(f'最小值点\n({x1_exact:.3f}, {g_min_exact:.3f})',
    
75.              xy=(x1_exact, g_min_exact),
    
76.              xytext=(x1_exact + 0.8, g_min_exact - 6),
    
77.              fontsize=10, color='purple',
    
78.              arrowprops=dict(arrowstyle='->', color='purple'))
    
79. 
    
80. ax1.plot(x2_exact, g_min_exact, 'mo', markersize=10, zorder=5)
    
81. ax1.annotate(f'({x2_exact:.3f}, {g_min_exact:.3f})',
    
82.              xy=(x2_exact, g_min_exact),
    
83.              xytext=(x2_exact + 0.3, g_min_exact - 8),
    
84.              fontsize=9, color='purple',
    
85.              arrowprops=dict(arrowstyle='->', color='purple'))
    
86. 
    
87. ax1.set_xlabel('x', fontsize=12)
    
88. ax1.set_ylabel('f(x)', fontsize=12)
    
89. ax1.set_title('原函数 f(x) 的图像', fontsize=13)
    
90. ax1.set_ylim(-25, 15)
    
91. ax1.legend(fontsize=9)
    
92. ax1.grid(True, alpha=0.3)
    
93. 
    
94. # ---- 图2：化简后的三次函数 g(u) ----
    
95. u = np.linspace(0, 8, 1000)
    
96. gu = g(u)
    
97. 
    
98. ax2 = axes[1]
    
99. ax2.plot(u, gu, color='darkorange', linewidth=2, label=r'$g(u)=(u-6.25)(u-2.25)(u-0.25)$')
    
100. ax2.axhline(0, color='black', linewidth=0.8, linestyle='--')
     
101. 
     
102. # 三个根
     
103. for r in [0.25, 2.25, 6.25]:
     
104.     ax2.plot(r, 0, 'ro', markersize=6)
     
105.     ax2.text(r, 1.5, f'{r}', ha='center', color='red', fontsize=9)
     
106. 
     
107. # 全局最小值
     
108. ax2.plot(u_min_exact, g_min_exact, 'mo', markersize=10, zorder=5)
     
109. ax2.annotate(f'全局最小\n(u*≈{u_min_exact:.3f}, {g_min_exact:.3f})',
     
110.              xy=(u_min_exact, g_min_exact),
     
111.              xytext=(u_min_exact + 0.5, g_min_exact + 3),
     
112.              fontsize=10, color='purple',
     
113.              arrowprops=dict(arrowstyle='->', color='purple'))
     
114. 
     
115. # 局部极大值（数值找）
     
116. mask = (u > 0.25) & (u < 2.25)
     
117. idx_max = np.argmax(gu[mask])
     
118. u_max = u[mask][idx_max]
     
119. g_max = gu[mask][idx_max]
     
120. ax2.plot(u_max, g_max, 'go', markersize=8)
     
121. ax2.annotate(f'局部极大\n({u_max:.3f}, {g_max:.3f})',
     
122.              xy=(u_max, g_max),
     
123.              xytext=(u_max + 0.8, g_max + 2),
     
124.              fontsize=9, color='green',
     
125.              arrowprops=dict(arrowstyle='->', color='green'))
     
126. 
     
127. ax2.set_xlabel('u = (x + 2.5)&#178;', fontsize=12)
     
128. ax2.set_ylabel('g(u)', fontsize=12)
     
129. ax2.set_title('化简后的三次函数 g(u) = (u-6.25)(u-2.25)(u-0.25)', fontsize=12)
     
130. ax2.legend(fontsize=9)
     
131. ax2.grid(True, alpha=0.3)
     
132. 
     
133. plt.tight_layout()
     
134. plt.savefig('min_solution.png', dpi=150)
     
135. plt.show()
     
136. 
     
137. print("\n图形已保存为 min_solution.png")

复制代码

Sora-空

0？

三体—夜航星

柯西不等式

137891

不会

I会成功

0

贝尔0703

应该是-12