{"ops":[{"insert":"高中生应该都见过下面这个不等式链\n"},{"attributes":{"class":"normal-img"},"insert":{"native-image":{"alt":"read-normal-img","url":"https://b2.sanwen.net/b_article/175df00c7e14168a949743985c65bd72fc8e7bee.jpg","width":2233,"height":2243,"size":378042,"status":"loaded"}}},{"insert":"本文将通过一个同构函数来证明这条不等式链\n先看一下这个函数（xi>0）\n\n\n"},{"attributes":{"class":"normal-img"},"insert":{"native-image":{"alt":"read-normal-img","url":"https://b2.sanwen.net/b_article/35043fb58d52d9d8b5ae3d3a9c1d9518e748a668.jpg","width":1079,"height":278,"size":34960,"status":"loaded"}}},{"insert":"观察他在0处的极限，我们会发现其形式和几何平均数（Gₙ）十分相似\n"},{"attributes":{"class":"normal-img"},"insert":{"native-image":{"alt":"read-normal-img","url":"https://b2.sanwen.net/b_article/34bc5800a9da8f2aacd923933e34b6a17fcb0356.jpg","width":2304,"height":2304,"size":585072,"status":"loaded"}}},{"insert":"而Hₙ Aₙ Qₙ则刚好是f（-1），f（1）和f（2）\n也即上面的不等式链等价于\n"},{"attributes":{"class":"normal-img"},"insert":{"native-image":{"alt":"read-normal-img","url":"https://b2.sanwen.net/b_article/cd2063e451202ca58e93e1d6e171c88b772981d3.jpg","width":1664,"height":347,"size":68007,"status":"loaded"}}},{"insert":"\n如此原先的不等式链就转化为了一个函数问题\n我们可以先画出函数的图象观察一下\n如图\n\n"},{"attributes":{"class":"normal-img"},"insert":{"native-image":{"alt":"read-normal-img","url":"https://b2.sanwen.net/b_article/6bf635fedf80052bcd343ea028c2892d4e5ec513.jpg","width":1079,"height":741,"size":89260,"status":"loaded"}}},{"insert":"基于这个图象，我们可以大胆猜想：f（x）是单调递增的（xi不全相等）"},{"attributes":{"header":1},"insert":"\n"},{"insert":"研究单调性，肯定先求导\n"},{"attributes":{"class":"normal-img"},"insert":{"native-image":{"alt":"read-normal-img","url":"https://b2.sanwen.net/b_article/3704fd5ff1a07ccf5bce74779a65736234708b94.jpg","width":1536,"height":555,"size":105010,"status":"loaded"}}},{"insert":"看起来比较长，但我们只需判断导数后半部分的正负性即可\n也即证明\n"},{"attributes":{"class":"normal-img"},"insert":{"native-image":{"alt":"read-normal-img","url":"https://b2.sanwen.net/b_article/95015e87ee757d1144aa908fd495ce272cafed9f.jpg","width":1536,"height":734,"size":144052,"status":"loaded"}}},{"insert":"下面就是证明过程了（可以先自己证一下）\n\n\n"},{"attributes":{"class":"normal-img"},"insert":{"native-image":{"alt":"read-normal-img","url":"https://b2.sanwen.net/b_article/17cbe7888e148427759c2edf62a0f63b87a01618.jpg","width":1280,"height":1214,"size":251815,"status":"loaded"}}},{"insert":"下面还有另一种解法【由B站“求导宗师的线性空间”（UID:2062338024）给出】\n"},{"attributes":{"class":"normal-img"},"insert":{"native-image":{"alt":"read-normal-img","url":"https://b2.sanwen.net/b_article/989ff2949a7185ef9b72c71f0a61a1ffb4d2642a.jpg","width":1280,"height":1026,"size":716437,"status":"loaded"}}},{"insert":"至此我们就成功证明了基本不等式链\n"}]}
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