{"ops":[{"insert":"高中生應該都見過下面這個不等式鏈\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":"本文將通過一個同構(gòu)函數(shù)來證明這條不等式鏈\n先看一下這個函數(shù)（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":"觀察他在0處的極限，我們會發(fā)現(xiàn)其形式和幾何平均數(shù)（G?）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H? A? Q?則剛好是f（-1），f（1）和f（2）\n也即上面的不等式鏈等價于\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如此原先的不等式鏈就轉(zhuǎn)化為了一個函數(shù)問題\n我們可以先畫出函數(shù)的圖象觀察一下\n如圖\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":"基于這個圖象，我們可以大膽猜想：f（x）是單調(diào)遞增的（xi不全相等）"},{"attributes":{"header":1},"insert":"\n"},{"insert":"研究單調(diào)性，肯定先求導\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":"看起來比較長，但我們只需判斷導數(shù)后半部分的正負性即可\n也即證明\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":"下面就是證明過程了（可以先自己證一下）\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":"下面還有另一種解法【由B站“求導宗師的線性空間”（UID:2062338024）給出】\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":"至此我們就成功證明了基本不等式鏈\n"}]}