http://data.artofproblemsolving.com/images/latex/c/5/6/c56422ed0d243550e3be28e0776030937e25d999.gif

而当http://data.artofproblemsolving.com/images/latex/6/2/9/62903ba4e68c47f4fa620a57c7c4458d8d12ed7d.gif显然成立。

即http://data.artofproblemsolving.com/images/latex/3/b/c/3bc293749aaabdf2d64020facbf1bc5a3c159d63.gif。

所以，当http://data.artofproblemsolving.com/images/latex/0/6/c/06caa24880496bc6c903c01a7b9caa83fe8a195e.gif成立。

   故当http://data.artofproblemsolving.com/images/latex/7/4/9/74900ba12430d4ceb5e1a240f98d3edb8a32e930.gif，

 

http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif都成立。

   因此满足条件的最小值为http://data.artofproblemsolving.com/images/latex/c/9/1/c91824801233f9ece4dbc85e25336a1b77c221cb.gif。

   也可以通过构造数列证明：http://data.artofproblemsolving.com/images/latex/0/6/c/06caa24880496bc6c903c01a7b9caa83fe8a195e.gif。

   证明：设http://data.artofproblemsolving.com/images/latex/9/0/3/903e28b93d34db82638c5f7ea6dfa1f00c8ece37.gif，

http://data.artofproblemsolving.com/images/latex/a/c/3/ac3eaf43a3aa04549d6a97327b8e49f623ffa0ba.gif

因为当http://data.artofproblemsolving.com/images/latex/d/2/4/d24742f926d9941b8bbb1c164f09acecaf8c8933.gif时，有

http://data.artofproblemsolving.com/images/latex/c/9/9/c99f9d082324c307eab06cd71e1c1b8a4b804c64.gif

http://data.artofproblemsolving.com/images/latex/f/3/c/f3cc4a7469aea6324bd5e9c959eb5a4841720541.gif

所以，，所以数列递增，因此，当http://data.artofproblemsolving.com/images/latex/d/2/4/d24742f926d9941b8bbb1c164f09acecaf8c8933.gif时有，http://data.artofproblemsolving.com/images/latex/c/a/6/ca66d66fb8a69db25e47c7d079751c7268b502ee.gif，

 

即http://data.artofproblemsolving.com/images/latex/c/a/6/ca66d66fb8a69db25e47c7d079751c7268b502ee.gif。

   （2）最值法，就是应用结论http://data.artofproblemsolving.com/images/latex/7/f/9/7f9b147a4da1db302640684cb5ed7ffea77ca1a3.gif或

http://data.artofproblemsolving.com/images/latex/5/6/b/56bf52b1cc66a4843e31fb88feeed0ebbab0c572.gif

处理不等式恒成立的方法。

   下面用最值法求出使http://data.artofproblemsolving.com/images/latex/8/6/f/86f7e437faa5a7fce15d1ddcb9eaeaea377667b8.gif的范围。

   当http://data.artofproblemsolving.com/images/latex/4/f/7/4f7d81389762366c131edcfc1778acc037b192e4.gif时， http://data.artofproblemsolving.com/images/latex/8/6/f/86f7e437faa5a7fce15d1ddcb9eaeaea377667b8.gif 可以为任意正实数；当http://data.artofproblemsolving.com/images/latex/4/d/0/4d040f015758516f8ca352dd8f6df8922f3759f5.gif；

   当http://data.artofproblemsolving.com/images/latex/5/1/1/5116c2e3ba6b187f826b55c207b649edf56a91d3.gif，则

http://data.artofproblemsolving.com/images/latex/1/7/7/177d87f3557eb1b619b7fac2705e520c50d21677.gif.

  下面求出当http://data.artofproblemsolving.com/images/latex/f/1/c/f1c2d165d7250f2fd4ebc5eeb10b8a20bcf8d131.gif的最大值。

 

   先证明:当http://data.artofproblemsolving.com/images/latex/4/7/7/4777e4bda3ab8a441ab13289cab8cb6e7e8c9479.gif时, http://data.artofproblemsolving.com/images/latex/4/4/f/44f9b5e21c8353d8282dcf93c2d3c6dd4c5442ff.gif 递减, 只须证，对http://data.artofproblemsolving.com/images/latex/4/7/7/4777e4bda3ab8a441ab13289cab8cb6e7e8c9479.gif，有，

 

即

http://data.artofproblemsolving.com/images/latex/0/b/1/0b1f5cd7ae0821fbb42b9685bf4c80f0c6a69d0b.gif

记http://data.artofproblemsolving.com/images/latex/6/3/d/63d886650c7e7c575c46b7ad3d2b6c4d3c72b939.gif (http://data.artofproblemsolving.com/images/latex/4/7/7/4777e4bda3ab8a441ab13289cab8cb6e7e8c9479.gif),，

当http://data.artofproblemsolving.com/images/latex/8/4/2/8420564c58024d24f42039a76f604befa9970349.gif，有

 

http://data.artofproblemsolving.com/images/latex/1/d/2/1d2875683c0508d9c929611de3c2121f1dcc1cef.gif
http://data.artofproblemsolving.com/images/latex/d/1/9/d1909dfa2664b1a4767455be0e7fd9623d01a5d8.gif

而当http://data.artofproblemsolving.com/images/latex/8/4/2/8420564c58024d24f42039a76f604befa9970349.gif时，有

http://data.artofproblemsolving.com/images/latex/1/4/7/14756e452ed8545cb825ceee2d923284fda0c3c3.gif

http://data.artofproblemsolving.com/images/latex/8/a/b/8ab3aded2cfccf33dbfeeb302827b27a6286cafe.gif

 

所以

http://data.artofproblemsolving.com/images/latex/7/8/f/78fd99eb6a3b94c4b2e459566fd9270e86572ddf.gif

http://data.artofproblemsolving.com/images/latex/c/3/a/c3af463c197837ca3a36632647c4b0e9bb0f786c.gif

因此

http://data.artofproblemsolving.com/images/latex/3/8/c/38c3b44d9d5ea5c310b633fe81c623088f000690.gif

即是递增数列，所以当http://data.artofproblemsolving.com/images/latex/4/7/7/4777e4bda3ab8a441ab13289cab8cb6e7e8c9479.gif时，有http://data.artofproblemsolving.com/images/latex/c/a/6/ca66d66fb8a69db25e47c7d079751c7268b502ee.gif，因此，对http://data.artofproblemsolving.com/images/latex/4/7/7/4777e4bda3ab8a441ab13289cab8cb6e7e8c9479.gif，有

 

，即http://data.artofproblemsolving.com/images/latex/4/4/f/44f9b5e21c8353d8282dcf93c2d3c6dd4c5442ff.gif递减，由此得到，当http://data.artofproblemsolving.com/images/latex/4/7/7/4777e4bda3ab8a441ab13289cab8cb6e7e8c9479.gif时，有。

 
所以，当时，对http://data.artofproblemsolving.com/images/latex/4/7/7/4777e4bda3ab8a441ab13289cab8cb6e7e8c9479.gif均有，；此时，对

 

http://data.artofproblemsolving.com/images/latex/0/6/c/06caa24880496bc6c903c01a7b9caa83fe8a195e.gif。

  故当http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif都成立成立。

 

  即当http://data.artofproblemsolving.com/images/latex/d/1/8/d1854cae891ec7b29161ccaf79a24b00c274bdaa.gif都成立。

(III). 因http://data.artofproblemsolving.com/images/latex/7/2/d/72dea082d929a17ebc17b683ba29ec017097eff8.gif，则

http://data.artofproblemsolving.com/images/latex/b/6/f/b6fa5d3836819d0fa2fa63761b0351814c5dbcdc.gif

http://data.artofproblemsolving.com/images/latex/c/9/b/c9b3f89f33709b94de91319c682f556f61e14338.gif

  先取特殊值探路：

  当http://data.artofproblemsolving.com/images/latex/9/2/e/92ee913214330816c21ed1490df01843b5cb2075.gif时，

http://data.artofproblemsolving.com/images/latex/6/b/f/6bfa7abce7a3083f1abe83cfcd885de97febc514.gif，

  当http://data.artofproblemsolving.com/images/latex/9/a/3/9a3240ad86b2b1d076825686aaf98565beeaac04.gif时，

http://data.artofproblemsolving.com/images/latex/7/a/1/7a1fadc7c4f9c5a66d0b9de53eca29407353e593.gif

http://data.artofproblemsolving.com/images/latex/6/6/d/66d8d8186fa7ed4c6f16049f4e1fd8cd1c5f49fb.gif http://data.artofproblemsolving.com/images/latex/b/f/7/bf7fb809c7b4cd053644a5390c3405ad7c2a1f4e.gif

令http://data.artofproblemsolving.com/images/latex/c/3/8/c3843eaf7ca5138b9b3474ccb7190feef3940669.gif，则

http://data.artofproblemsolving.com/images/latex/b/d/0/bd0f1e381b7dce1529171ac0b4556d6e3f801b52.gif

所以

http://data.artofproblemsolving.com/images/latex/5/2/e/52e6b1a44cc34a2aee0e03aebb92ebee4ef4aab5.gif

   故http://data.artofproblemsolving.com/images/latex/9/0/7/90790c2a8b5244d355544a298e98f72410ae0fe3.gif

   对于一般http://data.artofproblemsolving.com/images/latex/d/2/4/d24742f926d9941b8bbb1c164f09acecaf8c8933.gif的情形

http://data.artofproblemsolving.com/images/latex/d/c/8/dc8be9d9b7412442e0ae93dc6ffac671709310ef.gif

http://data.artofproblemsolving.com/images/latex/c/2/b/c2b6565923331ae14fc8ae871d16d0459d1cbd29.gif

http://data.artofproblemsolving.com/images/latex/8/c/1/8c1831ebcac1efb84b1ddc8dec2ee1fd7a380d2a.gif

在前面证明http://data.artofproblemsolving.com/images/latex/9/a/3/9a3240ad86b2b1d076825686aaf98565beeaac04.gif明，已证结论：

http://data.artofproblemsolving.com/images/latex/9/4/d/94d6eb424366bf157987346f9131416bb2b806a2.gif

及当http://data.artofproblemsolving.com/images/latex/5/1/c/51c6dd798ef5305391ba2e4955c2b860e27f212d.gif，有

http://data.artofproblemsolving.com/images/latex/4/2/9/4292a096b598e6e29d9e045b3be45f79aebe4dfc.gif

所以

http://data.artofproblemsolving.com/images/latex/a/1/b/a1be79743030299e59f3922839edeeb64b94b395.gif

即

http://data.artofproblemsolving.com/images/latex/e/5/a/e5a7972ba856f031387544acf2e52b0592d779dc.gif

实际上，我们可以证明稍强一点的结论：当http://data.artofproblemsolving.com/images/latex/0/c/0/0c0d7e220f4f43addc14f1e1211b8791a08a81c3.gif时，有

http://data.artofproblemsolving.com/images/latex/7/1/4/7143ca026b853fcddb3d1ae151d8f59f6c1f9ab2.gif

    (Ⅱ)、(III)两题的上述解法都是先从特殊入手，找出问题的结论或问题的解法。

 

其中(Ⅱ)题解法一用特殊值代入，猜出http://data.artofproblemsolving.com/images/latex/8/6/f/86f7e437faa5a7fce15d1ddcb9eaeaea377667b8.gif的最小值，再给出证明。而用数学归纳法给出

 

的证明，比参考答案用二项式定理给出的证明，思路自然且简单；(III)题先用等比数

 

列前http://data.artofproblemsolving.com/images/latex/6/1/1/611717e5a3b99149c591ce0194bb5df42da89b27.gif，然后通过特殊引路，找

到解一般情形的方法。需要注意的是http://data.artofproblemsolving.com/images/latex/0/2/6/0268a9e547c05923eb579abe60c7867e98469c5c.gif，逐

 

项比较http://data.artofproblemsolving.com/images/latex/b/9/e/b9eb2c8bb24d4cecbf516c96f7ad61fac5554863.gif ，有http://data.artofproblemsolving.com/images/latex/2/1/b/21b0ac9f4144d55cd9f4da0863bbb92994a2f986.gif，可以

 

按上述求差并分解因式证明得，也可以通过求导证得。

    另外，此题与2011年高考四川卷理科的压轴题：

已知函数http://data.artofproblemsolving.com/images/latex/2/3/f/23f7eab0854346e5d5fd26e5db22aa7b54cc4983.gif.

 

（Ⅰ）设函数http://data.artofproblemsolving.com/images/latex/6/1/7/617729eaf7470876a5410cad65a37f6af46517e1.gif的单调区间与极值；

 

（Ⅱ）设http://data.artofproblemsolving.com/images/latex/1/2/8/128646da94c305cb181250e34d58300d0fa0f87f.gif；

（Ⅲ）试比较http://data.artofproblemsolving.com/images/latex/a/0/6/a063853bdfd761520e06382373fe270b6078994b.gif]的大小.
（2011年高考数学四川卷理科第22题）

十分类似，两题第（Ⅰ）都涉及导数的应用，12年题是求切线的斜率，11题是求函数的

 

单调区间与极值；11年题第（Ⅱ）题是解含参数的方程，12年题第（Ⅱ）题是求不等式

 

恒成立的参数的范围；两题第（Ⅲ）题都是比较大小问题（实际上是证明不等式问题）。