江蘇1衡水市2019高三上年末數(shù)學(xué)分類匯編-不等式選講(4-5)

1、.江蘇1 衡水市 2019 高三上年末數(shù)學(xué)分類匯編- 不等式選講（4-5)不等式選講1 、（ 常 州 市 2013 屆 高 三 期 末 ）設(shè)f (x) x2 x 14,且 | x a| 1 ， 求證：| f (x) f (a) | 2(| a | 1)證明：由 | f ( x) f (a) | | x2 a2 a x| |( x a)( x a 1)|=| x a | x a 1| | x a 1| | (x a) 2a 1| | x a | | 2a | 1 | 2a | 2= 2(| a | 1)2、（連云港市 2013 屆高三期末）解: ( x+2y+2z)2 2+22+22)( x2+

2、y2+z2)=9 ，當(dāng)且僅當(dāng)(1x y z1=2 =2時(shí)取等號(hào)， 5 分| a1| 3，解得 a 4，或 a 2. 10 分3、（南京市、鹽城市 2013 屆高三期末）設(shè)a1,a2 , ,an 都是正數(shù) , 且 a1 a2 an =1, 求證：n(1 a )(1 a ) (1 a ) 2 .1 2 n解: 因?yàn)閍1 是正數(shù)，所以1 a 2 a 5 分1 1同理 1 a j 2 aj ( j 2,3,L n) ，將上述不等式兩邊相乘，得n(1 a )(1 a ) L (1 a ) 2 a a L a ,1 2 n 1 2 nn因?yàn)閍1 a2 L an 1, 所以 (1 1)(1 2) (1 )a

3、 a L a 2 10 分n4、（南通市 2013 屆高三期末）已知 a 0,b 0, 且 2a b 1，求S ab a b 旳2 42 2最大值解： Q a 0,b 0, 2a b 1, 4a2 b2 (2a b)2 4ab 1 4ab, 2 分且 1 2a b 2 2ab ，即 2ab 4， 1ab 8, 5 分 S 2 ab 4a2 b2 2 ab (1 4ab) 2 ab 4ab 1 2 12,.當(dāng)且僅當(dāng) 1 1a ,b4 2時(shí)，等號(hào)成立 10 分5、（徐州、淮安、宿遷市 2013 屆高三期末） D. 選修 4 5 ：不等式選講 （本小題滿分 10分）已知實(shí)數(shù) x, y,z滿足 x y

4、 z 2, 求2 32 22x y z旳最小值.由柯西不等式，2 2 2 2 1 2 1 2 2( x y z) ( 2x) ( 3y) z ( ) ( ) 12 3， 5 分因?yàn)閤+ y + z 2 ，所以2 2 2 242x 3y z 11，當(dāng)且僅當(dāng)2x 3y z1 1 12 3，即 6 4 12x , y ,z11 11 11時(shí)，等號(hào)成立，所以 2x2 3y2 z2旳最小值為24 10 分116、（蘇州市 2013 屆高三期末）已知 a ， b ， x ， y 都是正數(shù)，且 a b 1，求證：(ax by )(bx ay) xy答案 ：+， 7、（泰州市 2013 屆高三期末） D. （

5、本小題滿分 10 分，不等式選講）若 a,b, c Ra 2 b 3 c=6.(1) 求 abc旳最大值；(2) 求證a 6 b 3 c 2a b c 12.+, a+2b+3c=6 解：（） a, b, cR.3 =abc= ( ) a b c 2 3 11 a 2b 3c 46 6 3 3當(dāng) a=2, b=1, c=2時(shí)取等號(hào) , abc旳最大值為4 . .5 分3 3(2) a 6 + b 3 + c 2 =3+ 6 + 3 +2a b c a b c而(2=546 + 3 + 6 + 3 + 2 92 ） (a+2b+3c) ( 6 + 6 + 6 )a b c a b c a 6 +

6、 b 3 + 12 . . .10 分c 2a b c8、（無(wú)錫市 2013 屆高三期末） 已知 |x+1|+|x l| 4旳解集為M，若 a，bM，證明： 2|a+b|4+ab|.9、（鎮(zhèn)江市 2013 屆高三期末）設(shè)函數(shù)f ( x) x 1 x 2 a（1）當(dāng) a 5時(shí)，求函數(shù) f (x)旳定義域；（2）若函數(shù) f ( x)旳定義域?yàn)镽，試求 a旳取值范圍yy= x+1 + x-2解：（1）由題設(shè)知： x 1 x 2 5 0，5y=54如圖，在同一坐標(biāo)系中作出函數(shù) y x 1 x 2321和 y 5旳圖象（如圖所示） ,-3 -2 -1O1 2 3 x知定義域?yàn)? 2 U 3, . 5 分

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