ydwb 发表于 2026-9-28 08:05:30

求x(x+1)(x+2)(x+3)(x+4)(x+5)的最小值

求x(x+1)(x+2)(x+3)(x+4)(x+5)的最小值

爱吃榴莲 发表于 2026-9-28 12:18:49

不会

空python 发表于 2026-9-28 12:34:03

最小值可能是负数

ydwb 发表于 2026-9-28 18:16:07

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

ydwb 发表于 2026-9-29 08:09:16

import numpy as np
import matplotlib.pyplot as plt

# ============ 0. 中文字体配置 ============
plt.rcParams['font.sans-serif'] = ['SimHei', 'Microsoft YaHei', 'Arial Unicode MS']
plt.rcParams['axes.unicode_minus'] = False

# ============ 1. 定义原函数 f(x) ============
def f(x):
    return x * (x+1) * (x+2) * (x+3) * (x+4) * (x+5)

# ============ 2. 对称换元：t = x + 2.5, u = t^2 ============
def g(u):
    # g(u) = (u - 6.25)(u - 2.25)(u - 0.25)
    return (u - 6.25) * (u - 2.25) * (u - 0.25)

# ============ 3. 精确解析解 ============
# 对 g(u) 求导：g'(u) = 3u^2 - 17.5u + 16.1875 = 0
# 极小值点 u = (35 + 8*sqrt(7)) / 12
u_min_exact = (35 + 8*np.sqrt(7)) / 12
g_min_exact = (u_min_exact - 6.25) * (u_min_exact - 2.25) * (u_min_exact - 0.25)

t_min = np.sqrt(u_min_exact)
x1_exact = -2.5 - t_min
x2_exact = -2.5 + t_min

print("=" * 55)
print("【解析解】")
print("=" * 55)
print(f"极小值点 u* = (35 + 8√7)/12 ≈ {u_min_exact:.6f}")
print(f"最小值 g(u*) = (-160 - 112√7)/27 ≈ {g_min_exact:.6f}")
print(f"对应 t = ±√u* ≈ ±{t_min:.6f}")
print(f"对应 x1 = -2.5 - √u* ≈ {x1_exact:.6f}")
print(f"对应 x2 = -2.5 + √u* ≈ {x2_exact:.6f}")

# ============ 4. 数值验证 ============
print("\n" + "=" * 55)
print("【数值验证】")
print("=" * 55)
print(f"f(x1) = {f(x1_exact):.8f}")
print(f"f(x2) = {f(x2_exact):.8f}")
print(f"f(-2.5) = {f(-2.5):.8f}    (对称中心，非最小值)")
print(f"f(-4.66) = {f(-4.66):.8f}(近似点)")
print(f"f(-4.67) = {f(-4.67):.8f}")
print(f"f(-4.65) = {f(-4.65):.8f}")

# 用 scipy 数值求最小值做交叉验证
try:
    from scipy.optimize import minimize_scalar
    res = minimize_scalar(f, bounds=(-5, 0), method='bounded')
    print(f"\nscipy 数值最小值: x ≈ {res.x:.6f}, f(x) ≈ {res.fun:.8f}")
except ImportError:
    print("\n(未安装 scipy，跳过数值优化验证)")

# ============ 5. 图形 ============
fig, axes = plt.subplots(1, 2, figsize=(14, 5.5))

# ---- 图1：原函数 f(x) ----
x = np.linspace(-6, 1, 2000)
y = f(x)

ax1 = axes
ax1.plot(x, y, color='blue', linewidth=2, label=r'$f(x)=x(x+1)(x+2)(x+3)(x+4)(x+5)$')
ax1.axhline(0, color='black', linewidth=0.8, linestyle='--')
ax1.axvline(-2.5, color='gray', linewidth=0.8, linestyle=':', label='x = -2.5 (对称中心)')

# 标出整数根
for r in [-5, -4, -3, -2, -1, 0]:
    ax1.plot(r, 0, 'ro', markersize=5)
    ax1.text(r, 3, f'{r}', ha='center', color='red', fontsize=9)

# 标出最小值点
ax1.plot(x1_exact, g_min_exact, 'mo', markersize=10, zorder=5)
ax1.annotate(f'最小值点\n({x1_exact:.3f}, {g_min_exact:.3f})',
             xy=(x1_exact, g_min_exact),
             xytext=(x1_exact + 0.8, g_min_exact - 6),
             fontsize=10, color='purple',
             arrowprops=dict(arrowstyle='->', color='purple'))

ax1.plot(x2_exact, g_min_exact, 'mo', markersize=10, zorder=5)
ax1.annotate(f'({x2_exact:.3f}, {g_min_exact:.3f})',
             xy=(x2_exact, g_min_exact),
             xytext=(x2_exact + 0.3, g_min_exact - 8),
             fontsize=9, color='purple',
             arrowprops=dict(arrowstyle='->', color='purple'))

ax1.set_xlabel('x', fontsize=12)
ax1.set_ylabel('f(x)', fontsize=12)
ax1.set_title('原函数 f(x) 的图像', fontsize=13)
ax1.set_ylim(-25, 15)
ax1.legend(fontsize=9)
ax1.grid(True, alpha=0.3)

# ---- 图2：化简后的三次函数 g(u) ----
u = np.linspace(0, 8, 1000)
gu = g(u)

ax2 = axes
ax2.plot(u, gu, color='darkorange', linewidth=2, label=r'$g(u)=(u-6.25)(u-2.25)(u-0.25)$')
ax2.axhline(0, color='black', linewidth=0.8, linestyle='--')

# 三个根
for r in :
    ax2.plot(r, 0, 'ro', markersize=6)
    ax2.text(r, 1.5, f'{r}', ha='center', color='red', fontsize=9)

# 全局最小值
ax2.plot(u_min_exact, g_min_exact, 'mo', markersize=10, zorder=5)
ax2.annotate(f'全局最小\n(u*≈{u_min_exact:.3f}, {g_min_exact:.3f})',
             xy=(u_min_exact, g_min_exact),
             xytext=(u_min_exact + 0.5, g_min_exact + 3),
             fontsize=10, color='purple',
             arrowprops=dict(arrowstyle='->', color='purple'))

# 局部极大值（数值找）
mask = (u > 0.25) & (u < 2.25)
idx_max = np.argmax(gu)
u_max = u
g_max = gu
ax2.plot(u_max, g_max, 'go', markersize=8)
ax2.annotate(f'局部极大\n({u_max:.3f}, {g_max:.3f})',
             xy=(u_max, g_max),
             xytext=(u_max + 0.8, g_max + 2),
             fontsize=9, color='green',
             arrowprops=dict(arrowstyle='->', color='green'))

ax2.set_xlabel('u = (x + 2.5)²', fontsize=12)
ax2.set_ylabel('g(u)', fontsize=12)
ax2.set_title('化简后的三次函数 g(u) = (u-6.25)(u-2.25)(u-0.25)', fontsize=12)
ax2.legend(fontsize=9)
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('min_solution.png', dpi=150)
plt.show()

print("\n图形已保存为 min_solution.png")

Sora-空 发表于 2026-9-30 02:28:36

0？

三体—夜航星 发表于 2026-9-30 11:57:43

柯西不等式

137891 发表于 2026-9-30 20:51:42

不会

I会成功 发表于 2026-10-3 15:00:16

0

贝尔0703 发表于 6 天前

应该是-12
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