作业帮试比较a^3+8a与5a²+4的大小如题
来源：学生作业帮助网 编辑：作业帮 时间：2024/05/06 01:48:59
试比较a^3+8a与5a²+4的大小如题
试比较a^3+8a与5a²+4的大小
如题
试比较a^3+8a与5a²+4的大小如题
(a³ + 8a) - (5a² + 4)
= a³ - a² - 4(a² - 2a + 1)
= a²(a - 1) - 4(a - 1)²
= (a - 1)(a² - 4a + 4)
= (a - 1)(a - 2)²
当a=2,a³ + 8a = 5a² + 4
当a>1,a≠2,a³ + 8a > 5a² + 4
当a<1,a³ + 8a < 5a² + 4
(a³ + 8a) - (5a² + 4)
= a³ - a² - 4(a² - 2a + 1)
= a²(a - 1) - 4(a - 1)²
= (a - 1)(a² - 4a + 4)
= (a - 1)(a - 2)²
当(a - 1)(a - 2)²...
全部展开
(a³ + 8a) - (5a² + 4)
= a³ - a² - 4(a² - 2a + 1)
= a²(a - 1) - 4(a - 1)²
= (a - 1)(a² - 4a + 4)
= (a - 1)(a - 2)²
当(a - 1)(a - 2)²=0时 解得a=1,a=2 a³ + 8a = 5a² + 4
当(a - 1)(a - 2)²<0时 解得1
收起
a<=0时a^3+8a<5a²+4
a>0时a^3+8a-5a²-4=a^2(a-2)-3a(a-2)+2(a-2)
=(a-2)^2(a-1)
0
a>1,a^3+8a>5a²+4
故a<1时a^3+8a<5a²+4
a=1,a^3+8a=5a²+4
a>1,a^3+8a>5a²+4