问几道数学的不等式问题
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问几道数学的不等式问题
1.已知a<b<c且a+b+c=0,则b^2-4ab 0(填>或<或≤或≥)
2.函数y=log2[x^2/(x-1)]的值域为 (2为底,x^2/(x-1)为真数)
3.设a,b∈R+,求证a^a·b^b≥a^b·b^a(作商法)
1.已知a<b<c且a+b+c=0,则b^2-4ab 0(填>或<或≤或≥)
2.函数y=log2[x^2/(x-1)]的值域为 (2为底,x^2/(x-1)为真数)
3.设a,b∈R+,求证a^a·b^b≥a^b·b^a(作商法)
1、b^2-4ac=(-a-c)^2-4ac=a^2-2ac+c^2=(a-c)^2>0
2、x^2/(x-1)>0
x>1
设x-1=t>0,则x=1+t
y=log2[(t^2+2t+1)/t]=log2(t+1/t+2)≥log2(2+2)=2
值域为[2,+∞)
3、a^a·b^b/(a^b·b^a)
=(a/b)^a·(b/a)^b
=(a/b)^(a-b)
1)当a>b>0时,a/b>1,a-b>0
(a/b)^(a-b)>1
a^a·b^b>a^b·b^a
2)当a=b>0时,a/b=1,a-b=0
(a/b)^(a-b)=1
a^a·b^b=a^b·b^a
3)当0
2、x^2/(x-1)>0
x>1
设x-1=t>0,则x=1+t
y=log2[(t^2+2t+1)/t]=log2(t+1/t+2)≥log2(2+2)=2
值域为[2,+∞)
3、a^a·b^b/(a^b·b^a)
=(a/b)^a·(b/a)^b
=(a/b)^(a-b)
1)当a>b>0时,a/b>1,a-b>0
(a/b)^(a-b)>1
a^a·b^b>a^b·b^a
2)当a=b>0时,a/b=1,a-b=0
(a/b)^(a-b)=1
a^a·b^b=a^b·b^a
3)当0