设a>0,b>0,则下列不等式中正确的有几个()(1)a2+1>a;(2)(a+1a)(b+1b)≥4;(3)(a+b)(1a+1b)≥4;(4)a2+9>6a;(5)a2+1+1a2+1>2.A.1

题目简介

设a>0,b>0,则下列不等式中正确的有几个()(1)a2+1>a;(2)(a+1a)(b+1b)≥4;(3)(a+b)(1a+1b)≥4;(4)a2+9>6a;(5)a2+1+1a2+1>2.A.1

题目详情

设a>0,b>0,则下列不等式中正确的有几个(  )
(1)a2+1>a;
(2)(a+
1
a
)(b+
1
b
)≥4;
(3)(a+b)(
1
a
+
1
b
)≥4;
(4)a2+9>6a;
(5)a2+1+
1
a2+1
>2.
A.1B.2C.3D.4
题型:单选题难度:中档来源:不详

答案

∵a>0,b>0,
∴a2+1≥2a>a,
∴①正确;
(a+class="stub"1
a
)(b+class="stub"1
b
)=(ab+class="stub"1
ab
)+(class="stub"b
a
+class="stub"a
b
)≥2+2=4,等号在a=b时成立,
∴②正确;
(a+b)(class="stub"1
a
+class="stub"1
b
)=2+class="stub"b
a
+class="stub"a
b
≥4.等号在a=b时成立,
∴③正确;
∵a2+9-6a=(a-3)2≥0,
∴a2+9≥6a.等号在a=3时成立,
∴④错误;
a2+1+class="stub"1
a2+1
≥2.等号在a=0时成立,但a>0,
∴a2+1+class="stub"1
a2+1
>2,
∴⑤正确.
故正确的不等式有4个.
故选D.

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