福建省泉州市三校（铭选中学、泉州九中、+侨光中学）2022-2023学年高二下学期4月期中联考数学试题含答案.docx

<p>铭选中学、泉州九中、 侨光中学2023春季高二年期中联考数学试卷（考试时间：120 分钟 &nbsp; &nbsp; 满分：150 分 ）命题者：陈美珠 &nbsp;林公兴 &nbsp; &nbsp;审核者： 林公兴一、单选题：本题共8小题，每小题5分，共40分。在每个小题给出的选项中，只有一项是符合题目要求的。1记为等差数列的前项和若，则的公差为（&nbsp;&nbsp;&nbsp;&nbsp;）A1 &nbsp; &nbsp; &nbsp; &nbsp;B2 &nbsp; &nbsp; &nbsp; &nbsp; C4D82将5个1和2个0随机排成一行，则2个0不相邻的概率为（&nbsp;&nbsp;&nbsp;&nbsp;）ABCD3随机变量的分布列如下表，若，则（&nbsp;&nbsp;&nbsp;&nbsp;）A0B2C3D44函数的图象大致为（&nbsp;&nbsp;&nbsp;&nbsp;）ABCD5长时间玩手机可能影响视力，据调查，某校学生大约40%的人近视，而该校大约有20%的学生每天玩手机超过1，这些人的近视率约为50%.现从每天玩手机不超过1的学生中任意调查一名学生，则他近视的概率为（&nbsp;&nbsp;&nbsp;&nbsp;）ABCD6 在某互联网大会上，为了提升安保级别，将甲、乙等5名特警分配到3个不同的路口执勤，每个人只能分配到1个路口，每个路口最少1人，且甲和乙不能安排在同一个路口，则不同的安排方法有（&nbsp;&nbsp;&nbsp;&nbsp;）A180种B150种C96种D114种7 定义在R上的函数满足，是的导函数，且，则不等式（其中e为自然对数的底数）的解集为（&nbsp;&nbsp;&nbsp;&nbsp;）ABCD8已知函数，若的图像与轴有3个不同的交点，则实数的取值范围是ABCD二 多选题：本题共4小题，每小题5分，共20分。在每个小题给出的选项中，有多项符合题目要求，全部选对的5分，部分选对的2分，有选错的0分。9已知的二项展开式中二项式系数之和为64，下列结论正确的是（&nbsp;&nbsp;&nbsp;&nbsp;）A二项展开式中各项系数之和为B二项展开式中二项式系数最大的项为C二项展开式中无常数项D二项展开式中系数最大的项为10正方体的棱长为2，E，F，G分别为的中点，则（&nbsp;&nbsp;&nbsp;&nbsp;） A 直线与直线垂直 &nbsp; &nbsp; &nbsp; &nbsp; B直线与平面平行C平面截正方体所得的截面面积为 &nbsp; &nbsp; &nbsp;D点C到平面的距离为11 已知编号为1，2，3的三个盒子，其中1号盒子内装有两个1号球，一个2号球和一个3号球；2号盒子内装有两个1号球，一个3号球；3号盒子内装有三个1号球，两个2号球.若第一次先从1号盒子内随机抽取1个球，将取出的球放入与球同编号的盒子中，第二次从该盒子中任取一个球，则下列说法正确的是（&nbsp;&nbsp;&nbsp;&nbsp;）A在第一次抽到2号球的条件下，第二次抽到1号球的概率为B第一次抽到2号球且第二次抽到1号球的概率为C第二次抽到3号球的概率为D如果第二次抽到的是3号球，则它来自1号盒子的概率最大12已知函数，则下列结论正确的是（&nbsp;&nbsp;&nbsp;&nbsp;）A函数的图像关于原点对称B若在R上单调递增，则C当时，函数恰有两个零点D当时，函数恰有两个极值点三、填空题：本大题共4小题，每小题5分，共20分13的展开式中x的系数是_.14.在杨辉三角中，图中虚线上的数、.依次构成数列，则_（用数字作答）15.某集团第一年年初给下属企业甲制造厂投入生产资金万元，到年底资金增长了，以后每年资金年增长率与第一年相同.集团要求甲制造厂从投入生产资金开始，每年年底上缴资金万元，并将剩余资金全部投入下一年生产.设第年年底甲制造厂上缴资金后的剩余资金为万元，若，则正整数的最小值为_.（取，）16.已知函数，对任意的，恒成立，则的取值范围是_.四、解答题（本大题共6小题，共70分解答应写出文字说明，证明过程或演算步骤除17题为10分外，1822题均为12分）17.已知函数在处有极值，且曲线在点处的切线与直线平行(1)求；(2)若方程在区间上有三个不同的根，求的取值范围 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 18已知数列满足（）(1)求数列的通项公式：(2)设为数列的前项和19如图1，在菱形中，为的中点，.现将沿翻折至，并连接，得到如图2所示的四棱锥，且.(1)证明：；(2)在棱上是否存在点，使得与平面所成的角的正弦值为若存在，求出的值；若不存在，请说明理由.20已知数列的首项为1，令（1）若为常数列，求的解析式；（2）若是公比为3的等比数列，试求数列的前项和21 如图是飞行棋部分棋盘图示，飞机的初始位置为0号格。抛掷一个质地均匀的骰子，若拋出的点数为1，2，飞机在原地不动；若抛出的点数为3，4，飞机向前移一格；若抛出的点数为5，6，飞机向前移两格。记事件为“抛掷骰子一次后，飞机到达1号格”，记事件为“抛掷骰子两次后，飞机到达2号格”。(1)求；并判断事件是否独立，并说明理由；(2)抛掷骰子2次后，记飞机所在格子的号为，求随机变量的分布列和数学期望22已知函数(1)当时，讨论函数的单调性：(2)若函数恰有两个极值点且，求的最大值 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;铭选中学、泉州九中、 侨光中学2023春季高二年期中联考数学参考答案202304201-8:C D B B A D A A &nbsp; 9.ABD &nbsp; &nbsp;10.BCD &nbsp; &nbsp;11.AD &nbsp; &nbsp; 12.ABD13. &nbsp; &nbsp;14. 220 &nbsp; &nbsp;15.6 &nbsp; &nbsp;16.17.解：函数的导函数为， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;. 1分由题意得即，解得， &nbsp; &nbsp; &nbsp; &nbsp; . 3分， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;. 4分经检验，符合题意， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 5分；(2)解：由（1）得， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;. 6分 当或时， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 当时，所以函数在和上递增，在上递减， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; .7分 &nbsp;函数在处取得极大值，在处取极小值， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 8分 &nbsp;， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 9分 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;因为方程在区间上有三个不同的根，所以函数和函数在区间上有三个交点。 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 10分 &nbsp;18.（1）&nbsp;&nbsp;时，&nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 2分 - 得， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 3分 &nbsp;， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 4分 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 当时，满足上式。 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; .5分 &nbsp;故； &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 6分 &nbsp; ( 备注：未分类讨论即未体现，扣2分；未检验扣1分）（2）由（1）得：， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp;&nbsp;， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;. 7分 &nbsp;两边同乘以得：&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;. 8分 -得： &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; .9分 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;. 10分 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 11分 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;. 12分 19（1）证：在中，. 1分 由，得，所以，即在空间中. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 2分 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (证明为正，三线合一，同样得分) 又，所以.连接，在中， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 3分 (在中，用余弦定理求解，同样得分) 在中，由，得，所以. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 4分 又，平面，所以平面 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 5分 (未说明扣一分) 又平面，所以. &nbsp; &nbsp;. 6分 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; （2） 由（1）可知两两垂直，分别以所在直线为轴，轴，轴建立空间直角坐标， &nbsp; . 7分 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 则第1页 共 6页.设平面的法向量令，则. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 8分 假设棱上存在点，使得与平面所成的角的正弦值为，设，则， &nbsp; &nbsp; &nbsp; &nbsp; ， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;. 9分 所以，. 10分 整理得，解得 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 11分 又，所以，故. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;. 12分 20（1）由为常数列，又，所以， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;. 1分 所以 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;. 4分 （2）由是公比为3的等比数列，且，所以， &nbsp; &nbsp; . 5分 所以， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 6分 所以 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 7分 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 8分 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 9分 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;. 10分 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 11分 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 12分 第 3 页 共6 页21（1）由题意，, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;. 2分 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;. 3分（没有事件符号，仅有代数计算式扣2分）记“第i次抛掷骰子后，飞机向前移动两格”为事件“第i次抛掷骰子后，飞机在原地不动”为事件 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 4分由事件的互斥性和概率乘法公式得 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . 5分 &nbsp; （没有事件符号，关系，仅有代数计算式扣1分）因此，所以事件，相互独立 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; .6分 &nbsp; &nbsp; &nbsp;（由得出事件，相互独立，也得分）（2）随机变量的可能取值为0，1，2，3，4， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; .7分 ， &nbsp; &nbsp;， &nbsp; ， &nbsp; &nbsp;.9分 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;（概率值计算对两个給1分，全对給2分，只对一个不给分。没有列式过程，只有正确结果得1分）所以随机变量的分布列为01234 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; .10分 所以 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;.12分 （列式，得数各1分）第 4 页 共 6 页22 （1）解法一：（根据根是否存在和根与端点的大小比较分类讨论）函数的定义域为（0，+）， &nbsp; &nbsp; &nbsp; &nbsp; .1分 1)当a0时，在（0，+）恒成立，f（x）在（0，+）上单调递增； &nbsp; &nbsp; .2分 &nbsp;2)当0ae时，令，则设令,则x=lna &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; .3分&lt;1&gt;若0a1,则x=lna0 ,则 &gt;0在（0，+）恒成立在（0，+）单调递增,所以在（0，+）恒成立 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; . .4分&lt;2&gt;若1ae 时, &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 0xlna时，g（x）单调递减，xlna时，g（x）单调递增， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; g（x）g（lna）aalnaa（1lna）0， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ，在（0，+）上单调递增； &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;.5分 综上，当ae时，f（x）在（0，+）上单调递增。 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; .6分 （1） 解法二：（根据导数值恒同号，参数范围进行讨论）函数的定义域为（0，+）， &nbsp; &nbsp; &nbsp; &nbsp; .1分 由,令则 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;.2分 令则 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;.3分 当时 ,;当时 ,;当时 ,;所以函数在单调递减，在单调递增，当时， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; .4分 所以当时，在（0，+）恒成立，此时 .5分 当时，在（0，+）上单调递增。 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;.6分 （1）解法三：（根据导数值放缩，恒同号讨论）函数f（x）的定义域为（0，+）， &nbsp; &nbsp; &nbsp; &nbsp; .1分 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;3分 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 令 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;4分 当0x1时，g（x）单调递减，当x1时，g（x）单调递增， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; g（x）g（1）0 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; .5分 在（0，+）恒成立，在（0，+）上单调递增； &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 所以，当ae时，f（x）在（0，+）上单调递增； &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; .6分（2）依题意，则，所以 &nbsp; .7分 设，则t1，x2tx1， ， (各1分) &nbsp; ， .9分 设则， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;.10分 设，则，在（1，+）单调递增，则，则h（t）在（1，+）单调递增， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;又，即h（t）2ln3，h（3）2ln3， &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; .11分 t（1，3， 所以的最大值为3 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; .12分 第 6页 共6 页</p>