函数 高一 比较3个数字的大小

比较:2^(1/2) 和 （2/3)^(-1) 和3^(1/3)的 大小
令,m1=2^(1/2),m2=(2/3)^(-1),m3=3^(1/3)
log2(m1)=1/2,
log2/3(m2)=-1,
log2(m2)=(-1)[log2(2)-log2(3)=log2(3)-1,
log3(m3)=1/3,
log2(m3)=1/3log2(3).

log2(m2)-log2(m3)=log2(3)-1-1/3log2(3)
=2/3log2(3)-1=2/3[log2(3)-3/2log2(2)]
=2/3[log2(3/2√2)]
∵3/2√2＞1,∴log2(3/2√2)＞0,
log2(m2)-log2(m3)＞0,
log2(m2)＞log2(m3),

log2(m3)-log2(m1)=1/3log2(3)-1/2=1/3[log2(3)-3/2log2(2)]
=1/3[log2(3)-log2(2)^3/2]
=1/3[log2(3/2√2)]
∵3/2√2＞1,∴log2(3/2√2)＞0,
log2(m3)-log2(m1)＞0,
log2(m3)＞log2(m1).

∴（2/3)^(-1)＞3^(1/3)＞2^(1/2) .

2^(1/2)=2^(3/6)=8^(1/6)<1,
(2/3)^(-1)=(3/2)^1>1,
3^(1/3)=3^(2/6)=9^(1/6)
2^(1/2)<3^(1/3)<（2/3)^(-1)

2^(1/2)=2^(3/6)=8^(1/6)<1,
(2/3)^(-1)=(3/2)^1>1,
3^(1/3)=3^(2/6)=9^(1/6) <1
且 8^(1/6)<9^(1/6)
所以
2^(1/2)<3^(1/3)<（2/3)^(-1)

2^(1/2)=根号2，常识1.414
（2/3)^