基本不等式
1. 若![](https://wkrtcs.bdimg.com/rtcs/image?w=12&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"12","h":"19","dataType":"wmf","c":"word/media/image1.png","imgOriW":24,"imgOriH":38})
，下列不等式恒成立的是 （ ）
A．![](https://wkrtcs.bdimg.com/rtcs/image?w=57&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"57","h":"20","dataType":"wmf","c":"word/media/image3.png","imgOriW":114,"imgOriH":40})
B． C． D．
2. 若![](https://wkrtcs.bdimg.com/rtcs/image?w=53&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"53","h":"17","dataType":"wmf","c":"word/media/image7.png","imgOriW":106,"imgOriH":34})
且，则下列四个数中最大的是 （ ）
Ａ．![](https://wkrtcs.bdimg.com/rtcs/image?w=15&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"15","h":"37","dataType":"wmf","c":"word/media/image9.png","imgOriW":30,"imgOriH":74})
Ｂ． Ｃ．2ab Ｄ．a
3. 设x>0,则![](https://wkrtcs.bdimg.com/rtcs/image?w=83&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"83","h":"37","dataType":"wmf","c":"word/media/image11.png","imgOriW":166,"imgOriH":74})
的最大值为 （ ）
Ａ．3 Ｂ．![](https://wkrtcs.bdimg.com/rtcs/image?w=48&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"48","h":"21","dataType":"wmf","c":"word/media/image12.png","imgOriW":96,"imgOriH":42})
Ｃ． Ｄ．－1
4. 设![](https://wkrtcs.bdimg.com/rtcs/image?w=183&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"183","h":"23","dataType":"wmf","c":"word/media/image15.png","imgOriW":366,"imgOriH":46})
的最小值是( )
A. 10 B. ![](https://wkrtcs.bdimg.com/rtcs/image?w=29&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"29","h":"21","dataType":"wmf","c":"word/media/image16.png","imgOriW":58,"imgOriH":42})
C. D.
5. 若x, y是正数，且![](https://wkrtcs.bdimg.com/rtcs/image?w=56&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"56","h":"40","dataType":"wmf","c":"word/media/image19.png","imgOriW":112,"imgOriH":80})
，则xy有 （ ）
Ａ．最大值16 Ｂ．最小值![](https://wkrtcs.bdimg.com/rtcs/image?w=20&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"20","h":"37","dataType":"wmf","c":"word/media/image20.png","imgOriW":40,"imgOriH":74})
Ｃ．最小值16 Ｄ．最大值
6. 若a, b, c∈R，且ab+bc+ca=1, 则下列不等式成立的是 （ ）
A．![](https://wkrtcs.bdimg.com/rtcs/image?w=92&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"92","h":"20","dataType":"wmf","c":"word/media/image21.png","imgOriW":184,"imgOriH":40})
B．
C．![](https://wkrtcs.bdimg.com/rtcs/image?w=97&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"97","h":"41","dataType":"wmf","c":"word/media/image23.png","imgOriW":194,"imgOriH":82})
D．
7. 若x>0, y>0,且x+y![](https://wkrtcs.bdimg.com/rtcs/image?w=13&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"13","h":"16","dataType":"wmf","c":"word/media/image25.png","imgOriW":26,"imgOriH":32})
4,则下列不等式中恒成立的是 （ ）
A．![](https://wkrtcs.bdimg.com/rtcs/image?w=59&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"59","h":"40","dataType":"wmf","c":"word/media/image26.png","imgOriW":118,"imgOriH":80})
B． C． D．
8. a,b是正数，则![](https://wkrtcs.bdimg.com/rtcs/image?w=112&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"112","h":"37","dataType":"wmf","c":"word/media/image30.png","imgOriW":224,"imgOriH":74})
三个数的大小顺序是 （ ）
Ａ．![](https://wkrtcs.bdimg.com/rtcs/image?w=117&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"117","h":"37","dataType":"wmf","c":"word/media/image31.png","imgOriW":234,"imgOriH":74})
Ｂ．
Ｃ．![](https://wkrtcs.bdimg.com/rtcs/image?w=117&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"117","h":"37","dataType":"wmf","c":"word/media/image33.png","imgOriW":234,"imgOriH":74})
Ｄ．
9. 某产品的产量第一年的增长率为p，第二年的增长率为q，设这两年平均增长率为x，则有（ ）
Ａ．![](https://wkrtcs.bdimg.com/rtcs/image?w=59&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"59","h":"37","dataType":"wmf","c":"word/media/image35.png","imgOriW":118,"imgOriH":74})
Ｂ． Ｃ． Ｄ．
10. 下列函数中，最小值为4的是 （ ）
Ａ．![](https://wkrtcs.bdimg.com/rtcs/image?w=57&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"57","h":"37","dataType":"wmf","c":"word/media/image39.png","imgOriW":114,"imgOriH":74})
Ｂ．
Ｃ．![](https://wkrtcs.bdimg.com/rtcs/image?w=76&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"76","h":"23","dataType":"wmf","c":"word/media/image42.png","imgOriW":152,"imgOriH":46})
Ｄ．
11. 函数![](https://wkrtcs.bdimg.com/rtcs/image?w=76&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"76","h":"25","dataType":"wmf","c":"word/media/image44.png","imgOriW":152,"imgOriH":50})
的最大值为 .
12. 建造一个容积为18m3, 深为2m的长方形无盖水池，如果池底和池壁每m2 的造价为200元和150元，那么池的最低造价为 元.
13. 若直角三角形斜边长是1，则其内切圆半径的最大值是 .
14. 若x, y为非零实数，代数式![](https://wkrtcs.bdimg.com/rtcs/image?w=140&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"140","h":"41","dataType":"wmf","c":"word/media/image45.png","imgOriW":280,"imgOriH":82})
的值恒为正，对吗？答 .
三、解答题, 本大题共4小题，每小题12分，共48分，解答应写出必要的文字说明、证明过程和演算步骤.
15. 已知：![](https://wkrtcs.bdimg.com/rtcs/image?w=192&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"192","h":"23","dataType":"wmf","c":"word/media/image46.png","imgOriW":384,"imgOriH":46})
, 求mx+ny的最大值.
16. 设a, b, c![](https://wkrtcs.bdimg.com/rtcs/image?w=65&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"65","h":"21","dataType":"wmf","c":"word/media/image47.png","imgOriW":130,"imgOriH":42})
且a+b+c=1，求证：
17. 已知正数a, b满足a+b=1（1）求ab的取值范围;（2）求![](https://wkrtcs.bdimg.com/rtcs/image?w=49&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"49","h":"37","dataType":"wmf","c":"word/media/image49.png","imgOriW":98,"imgOriH":74})
的最小值.
18. 是否存在常数c,使得不等式![](https://wkrtcs.bdimg.com/rtcs/image?w=251&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"251","h":"44","dataType":"wmf","c":"word/media/image50.png","imgOriW":502,"imgOriH":88})
对任意正数x, y恒成立？试证明你的结论.
《基本不等式》综合检测
一、选择题
题号 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
答案 | A | B | C | D | C | A | B | C | C | C |
二．填空题
11. ![](https://wkrtcs.bdimg.com/rtcs/image?w=15&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"15","h":"37","dataType":"wmf","c":"word/media/image51.png","imgOriW":30,"imgOriH":74})
12.3600 13. 14.对
三、解答题
15．![](https://wkrtcs.bdimg.com/rtcs/image?w=29&md5sum=a30a295c62f88380abf03d0df0c99a87&sign=07482bb811&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=46&ipr={"t":"img","w":"29","h":"21","dataType":"wmf","c":"word/media/image53.png","imgOriW":58,"imgOriH":42})
16. 略 17. (1) (2) 18．存在，
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