基本不等式(高一新教材A版必修第一册)

8/r/n8/r/n基本不等式/r/n

/r/n第/r/n1/r/n课时基本不等式/r/n自/r/n主预习❶播/r/n新/r/ni/r/n：/r/n£iiHUYLJXlTAINXIM^HL/r/nI/r/n新知初探」/r/n重要不等式/r/na/r/n,/r/nb/r/n^/r/nR/r/n,/r/n有/r/na/r/n2/r/n+/r/nb/r/n2/r/n±/r/n2/r/nab/r/n，当且仅当/r/na/r/n=/r/nb/r/n时，等号成立./r/n基本不等式/r/n(1)/r/n有关概念：当/r/na/r/n,/r/nb/r/n均为正数时，把字叫做正数/r/na/r/n,/r/nb/r/n的算术平均数，把/r/nx/Ob/r/n叫做正数/r/na/r/n,/r/nb/r/n的几/r/n何平均数./r/n(2)/r/n不等式：当/r/na/r/n,/r/nb/r/n是任意正实数时，/r/na/r/n,/r/nb/r/n的几何平均数不大于它们的算术平均数，即/r/n価/r/nW/r/n孚,/r/n当且仅当/r/na/r/n=/r/nb/r/n时，等号成立./r/n匚昭］越身手/r/n_J/r/n1/r/n.不等式/r/na/r/n2/r/n+/r/n1/r/n±/r/n2/r/na/r/n中等号成立的条件是/r/n()/r/nA/r/n./r/na/r/n=/r/n±1/r/nB/r/nA/r/n./r/na/r/n=/r/n±1/r/nC/r/n./r/na/r/n=－/r/n1/r/nD/r/n./r/na/r/n=/r/n0/r/nB/r/n[/r/n当/r/na/r/n2/r/n+/r/n1/r/n=/r/n2/r/na/r/n,/r/n即/r/n(/r/na/r/n—/r/n1/r/n)/r/n2/r/n=/r/n0/r/n即/r/na/r/n=/r/n1/r/n时/r/n，“/r/n=/r/n"/r/n成立/r/n./r/n]/r/n2/r/n.已知/r/na/r/n,/r/nb/r/ne/r/n(/r/n0,1)/r/n,/r/n且/r/na/r/nH/r/nb/r/n，下列各式中最大的是/r/n(/r/n)/r/nB/r/n./r/n2pab/r/nD/r/n./r/na/r/n+/r/nb/r/nB/r/n./r/n2pab/r/nD/r/n./r/na/r/n+/r/nb/r/nC/r/n./r/n2/r/nab/r/nD/r/n[/r/nV/r/na/r/n,/r/nb/r/nG/r/n(0,1)/r/n,・°・/r/na/r/n2/r/nVa/r/n,/r/nb/r/n2/r/n<b/r/n,/r/n・/r/n°/r/n・/r/na/r/n2/r/n+/r/nb/r/n2/r/nVa/r/n+/r/nb/r/n,/r/n又/r/na/r/n2/r/n+/r/nb/r/n2/r/n>/r/n2/r/nab/r/n(/r/nV/r/na/r/nH/r/nb/r/n)/r/n,/r/n・/r/n2/r/nabVa/r/n2/r/n+b/r/n2/r/nVa+b/r/n./r/n又/r/nV/r/na+b>/r/n2^/r/nab/r/n(/r/nV/r/na/r/n^/r/nb/r/n)/r/n,・/r/na+b/r/n最大/r/n./r/n]/r/n3/r/n.已知/r/nab/r/n=/r/n1/r/n,/r/na/r/n>0/r/n,/r/nb/r/n>0/r/n,/r/n则/r/na/r/n+/r/nb/r/n的最小值为/r/n(/r/n)/r/nA/r/n./r/n1/r/nB/r/n./r/n2/r/nC/r/n./r/n4/r/nD/r/n./r/n8/r/nB/r/n[/r/nV/r/na/r/n>0/r/n,/r/nb/r/n>0/r/n,・・・/r/na/r/n+/r/nb/r/n±/r/n2/r/n、//r/nab/r/n=/r/n2/r/n,/r/n当且仅当/r/na=b=/r/n1/r/n时取等号/r/n，/r/n故/r/na/r/n+/r/nb/r/n的最小值为/r/n2.]/r/n

/r/n4/r/n.当/r/na/r/n,/r/nb/r/ne/r/nR/r/n时，下列不等关系成立的是/r/n./r/n①"/r/n寸"/r/nAj/r/nab/r/n:②/r/na/r/n—/r/nb/r/n三/r/n2/r/njab/r/n:③/r/na/r/n2/r/n+/r/nb/r/n2/r/n±/r/n2/r/nab/r/n:④/r/na/r/n2/r/n—/r/nb/r/n2/r/n±/r/n2/r/nab/r/n./r/n③/r/n［/r/n根据耳艺/r/nA/r/nxy/r/n，/r/n三剧/r/n成立的条件判断/r/n，/r/n知/r/n①②④/r/n错/r/n，/r/n只有/r/n③/r/n正确/r/n./r/n］/r/n合作按究堆奏弄/r/nEL£2UO1/r/nJ/r/nANliU/r/nJ'L/r/nSUYAN/r/n对基本不等式的理解/r/n例/r/n1/r/n】/r/n对基本不等式的理解/r/n例/r/n1/r/n】/r/n给出下面四个推导过程：/r/n①/r/n・・・/r/na/r/n、/r/nb/r/n为正实数，猪+餡/r/n2/r/n\/r/n：/r/n吟/r/nA/r/n2/r/n4/r/n丿/r/n4/r/n②/r/n・/r/n4/r/n丿/r/n4/r/n②/r/n・/r/n./r/n・/r/na/r/ne/r/nR/r/n,/r/na/r/nH/r/n0/r/n，・・・/r/na/r/n+/r/na/r/n±/r/n2/r/n\/r/n；；•/r/n心/r/n4/r/n；/r/n③/r/n・・・/r/nx/r/n、/r/ny/r/ne/r/nR/r/n，/r/nxy/r/n</r/n0/r/n，・/r/ny/r/n+/r/nx/r/n其中正确的推导为/r/n（/r/n）/r/n+/r/n(/r/n-/r/nX/r/n)l/r/n<-/r/n2/r/nA/r/n．/r/n①②/r/nB/r/n./r/n①③/r/nA/r/n．/r/n①②/r/nB/r/n./r/n①③/r/nC/r/n．/r/n②③/r/nD/r/n./r/n①②③/r/nba/r/n［/r/n①/r/n°・°/r/na/r/n、/r/nb/r/n为正实数，•：/r/nQ/r/n万为正实数/r/n，/r/n符合基本不等式的条件/r/n，/r/n故/r/n①/r/n的推导正确./r/nT/r/na/r/ne/r/nR/r/n，/r/na/r/nH/r/nO/r/n，/r/n不符合基本不等式的条件/r/n，/r/n4/r/n~/r/n-/r/n+/r/na/r/n±/r/n2:/r/n；-・/r/na/r/n=/r/n4/r/n是错误的/r/n./r/na/r/na/r/n由/r/nxy/r/nvo/r/n,/r/n得/r/n2/r/n》均为负数/r/n，/r/n但在推导过程中将整体/r/nx/r/n+/r/ny/r/n提出负号后/r/n，/r/n［/r/n―打、/r/n［/r/n―/r/nX/r/n〕/r/n均变为正数/r/n，/r/n符/r/nyx/r/nyx/r/nk/r/nyj\x/r/n丿/r/n合均值不等式的条件/r/n，/r/n故/r/n③/r/n正确/r/n./r/n］/r/n1/r/n./r/n基本不等式/r/n\/r/n1/r/n./r/n基本不等式/r/n\/r/n'ab^—2~/r/n（/r/na/r/n>/r/n0/r/n，/r/nb/r/n>/r/n0）/r/n反映了两个正数的和与积之间的关系/r/n./r/n2/r/n./r/n对基本不等式的准确掌握要抓住以下两个方面：/r/n（/r/n1）/r/n定理成立的条件是/r/na/r/n、/r/nb/r/n都是正数/r/n./r/n（/r/n2）/r/n“/r/n当且/r/n仅当/r/n"/r/n的含义：当/r/na=b/r/n时/r/n，/r/n，\/ObW^2^/r/n的等号成立/r/n，/r/n即/r/na=b^^2~/r/n=\//r/n0b/r/n;/r/n仅当/r/na=b/r/n时，空/三、/r/n'Ob/r/n的等号成立/r/n，/r/n即/r/na/r/n的等号成立/r/n，/r/n即/r/na/r/n+/r/nb/r/n~2~/r/n=/r/n\/r/n：/r/naba/r/n=/r/nb/r/n./r/n跟踪训练/r/n1/r/n．下列不等式的推导过程正确的是/r/n①若/r/nx/r/n>/r/nl/r/n,/r/n则/r/nx/r/n+/r/nx/r/n±/r/n2\/r/n：/r/n'x/r/n・/r/nx/r/n=/r/n2./r/nx/r/nx/r/n(/r/n-/r/n(/r/n-/r/n兀/r/n)/r/n+/r/n[/r/n-/r/nX/r/n]]/r/n②若/r/nx/r/n</r/n0/r/n,/r/n则/r/nx/r/n"-=一/r/nx/r/nW—/r/n2\/r/n」(/r/n一/r/nx/r/n)・/r/nb,/r/na/r/nba/r/n③若/r/na/r/n，/r/nbwR,/r/n则/r/na/r/n+/r/nb/r/n±/r/n2/r/n・/r/nab/r/n=/r/n2/r/n②/r/n[/r/n①/r/n中忽视了基本不等式等号成立的条件/r/n，/r/n当/r/nx/r/n=/r/n丄时即/r/nx/r/n=/r/n1/r/n时/r/n，/r/nx/r/n+~/r/n^/r/n2/r/n等号成立/r/n，/r/n因为/r/nx/r/n>/r/n1,/r/nxx/r/n所以/r/nx/r/n+/r/n1/r/n>/r/n2/r/n,/r/n③/r/n中忽视了利用基本不等式时每一项必须为正数这一条件/r/n./r/n]/r/nx/r/n利用基本不等式比较大小/r/n兴空/r/n2/r/n【例/r/n2/r/n】/r/n(1)/r/n已知/r/na/r/n,/r/nb/r/nG/r/nR/r/n+/r/n则下列各式中不一定成立的是/r/n()/r/nba/r/nA/r/n./r/na/r/n+/r/nb/r/n±/r/n2pab/r/nA/r/n成立;/r/n..a/r/n2/r/n+/r/nb/r/n2/r/nA/r/n成立;/r/n..a/r/n2/r/n+/r/nb/r/n2/r/nab/r/n=/r/n2\/r/n：/r/nab/r/n,・°・/r/nC/r/n成立;/r/n・/r/n°/r/n・/r/nD/r/n不一定成立/r/n./r/n(2)/r/n已知/r/na/r/n,/r/nb/r/n,/r/nc/r/n是两两不等的实数，则/r/np/r/n=/r/na/r/n2/r/n+/r/nb/r/n2/r/n+/r/nc/r/n2/r/n与/r/nq/r/n=/r/nab/r/n+/r/nbc/r/n+/r/nca/r/n的大小关系是/r/nD/r/n(/r/n2/r/n)/r/na/r/n2/r/n+/r/nb/r/n2/r/n+/r/nc/r/n2/r/n>/r/nab/r/n+/r/nbc/r/n+/r/nac/r/n[(1)/r/n由/r/n(2)/r/nT/r/na/r/n、/r/nb/r/n、/r/nc/r/n互不相等/r/n,/r/n・/r/na/r/n2/r/n+/r/nb/r/n2/r/n>/r/n2/r/nab/r/n，/r/nb/r/n2/r/n+/r/nc/r/n2/r/n>2/r/nac/r/n，/r/na/r/n2/r/n+/r/nc/r/n2/r/n>2/r/nac/r/n./r/n・/r/n2(/r/na/r/n2/r/n+/r/nb/r/n2/r/n+/r/nc/r/n2/r/n)>2(/r/nab/r/n+/r/nbc/r/n+/r/nac/r/n)/r/n./r/n即/r/na/r/n2/r/n+/r/nb/r/n2/r/n+/r/nc/r/n2/r/n>/r/nab/r/n+/r/nbc/r/n+/r/nac/r/n.]/r/n

/r/n1/r/n．/r/n在理解基本不等式时/r/n，/r/n要从形式到内含中理解/r/n，/r/n特别要关注条件/r/n．/r/n2/r/n./r/n运用基本不等式比较大小时应注意成立的条件/r/n，/r/n即/r/na/r/n+/r/nb/r/n±/r/n2\Gb/r/n成立的条件是/r/na/r/n>0/r/n,/r/nb/r/n>0/r/n,/r/n等号成立的条件是/r/na=b；a/r/n2/r/n+b/r/n2/r/n三/r/n2/r/nab/r/n成立的条件是/r/na/r/n,/r/nb/r/n^R,/r/n等号成立的条件是/r/na=b/r/n./r/n跟踪训练/r/n2/r/n./r/n女口果/r/n0/r/n</r/n跟踪训练/r/n2/r/n./r/n女口果/r/n0/r/n</r/na/r/n</r/nb/r/n</r/n1/r/n,/r/nP/r/n=/r/na/r/n+/r/nb/r/n

/r/n~T~/r/nQ=.jOb/r/n,/r/nM=\^+/r/n，那么/r/nP/r/n,/r/nQ/r/n,/r/nM/r/n的大小顺序是/r/n（/r/nA/r/n./r/nP/r/n〉/r/nQ/r/n〉/r/nM/r/nB/r/n./r/nM/r/n〉/r/nP/r/n〉/r/nQ/r/nC/r/n./r/nQ/r/n〉/r/nM/r/n〉/r/nP/r/nD/r/n./r/nM/r/n〉/r/nQ/r/n〉/r/nP/r/nB/r/n［/r/n显然竺笋/r/n>、莎,/r/n又因为兰笋/r/n（/r/n由/r/na/r/n+/r/nb/r/n>/r/n（/r/na/r/n~/r/n4/r/nb/r/n"/r/n也就是夕孑/r/nV/r/nI/r/n可得/r/n）/r/n，所以話再/r/n>価.故/r/nM/r/n>/r/nP/r/n>/r/nQ/r/n.］/r/n利用基本不等式证明不等式/r/n【例/r/n3/r/n】/r/n已知/r/na/r/n,/r/nb/r/n,/r/nc/r/n是互不相等的正数，且/r/na/r/n+/r/nb/r/n+/r/nc/r/n=/r/n1/r/n,/r/n求证：/r/n1/r/n+/r/n|/r/n+/r/n1/r/n>9./r/nabc/r/n［/r/n思路点拨/r/n］/r/n看到/r/n1/r/n+/r/n1/r/n+/r/n1>9/r/n,/r/n想到将/r/n“/r/n1/r/n”/r/n换成/r/n“/r/na/r/n+/r/nb/r/n+/r/nc/r/n”，/r/n裂项构造基本不等式的形式/r/n，/r/n用基本/r/n不等式证明/r/n./r/n［/r/n证明/r/n］/r/n'：/r/na/r/n,/r/nb/r/n,/r/nc/r/n^/r/nR/r/n+/r/n且/r/na/r/n+/r/nb/r/n+/r/nc/r/n=/r/n1/r/n,/r/n./r/n1/r/n(/r/n1/r/ni/r/n1/r/na+b+c|a+b+c/r/n(/r/na+b+c/r/n••a/r/n1/r/nb/r/n—++/r/nc/r/na/r/nb/r/nc/r/n=/r/n3/r/n+/r/nb/r/n+/r/nc/r/n+/r/na/r/n+/r/nb/r/n+/r/na/r/n+/r/naabbc/r/n=/r/n3/r/n+/r/nb/r/n+/r/nf/r/n)/r/n=/r/n3/r/n+/r/nb/r/n+/r/nf/r/n)/r/n+/r/n(/r/na/r/n+/r/na/r/n)/r/n+/r/n(b/r/n+/r/nb]/r/n±/r/n3/r/n+/r/n2/r/n=/r/n3/r/n+/r/n2/r/n+/r/n2/r/n+/r/n2/r/n当且仅当/r/na/r/n=/r/nb/r/n=/r/nc/r/n时取等号/r/n,/r/n•.-/r/n+/r/nb/r/n+/r/n->9./r/nabc/r/n母题探究］/r/n本例条件不变，求证：/r/n［/r/n十/r/n―/r/n1/r/n需—/r/nl/r/nj/r/n>8./r/n［/r/n证明/r/n］/r/nV/r/na/r/n,/r/nb/r/n,/r/nc/r/n^/r/nR/r/n+/r/n，/r/n且/r/na/r/n+/r/nb/r/n+/r/nc/r/n—/r/n1/r/n,/r/n1/r/nb+c/r/n1/r/na+c/r/n1/r/na+b/r/n•••aT/r/n—/r/n〒>°/r/n，/r/nbT/r/n—/r/nT~/r/n>/r/n0/r/n,/r/nc/r/n_/r/n1/r/n—/r/nb/r/n+/r/nca/r/n+/r/nca/r/n+/r/nb/r/nabc/r/n、/r/n2/r/n弋/r/nbc2pac・/r/n2/r/n弋/r/nab/r/n三/r/n—/r/n8/r/n,/r/nabc/r/n当且仅当/r/na/r/n—/r/nb/r/n—/r/nc/r/n时取等号,/r/n条件不等式的证明/r/n，/r/n要将待证不等式与已知条件结合起来考虑/r/n，/r/n比如本题通过/r/n“/r/n1/r/n”/r/n的代换/r/n，/r/n将不等式的左边化成齐次式/r/n,/r/n一方面为使用基本不等式创造条件/r/n,/r/n另一方面可实现约分与不等式的右边建立联系/r/n./r/n先局部运用基本不等式/r/n,/r/n再利用不等式的性质/r/n(/r/n注意限制条件/r/n)/r/n,/r/n通过相加/r/n(/r/n乘/r/n)/r/n合成为待证的不等式既是运用基本不等式时的一种重要技能/r/n,/r/n也是证明不等式时的一种常用方法/r/n./r/n㊃/r/n跟蹋训练/r/n已知/r/na/r/n,/r/nb/r/n,/r/nc/r/n^/r/nR/r/n，求证：/r/na/r/n4/r/n+/r/nb/r/n4/r/n+/r/nc/r/n4/r/n±/r/na/r/n2/r/nb/r/n2/r/n+/r/nb/r/n2/r/nc/r/n2/r/n+/r/nc/r/n2/r/na/r/n2/r/n./r/n［/r/n证明/r/n］/r/n由基本不等式可得/r/na/r/n4/r/n+/r/nb/r/n4/r/n—/r/n(/r/na/r/n2/r/n)/r/n2/r/n+/r/n(/r/nb/r/n2/r/n)/r/n2/r/n三/r/n2/r/na/r/n2/r/nb/r/n2/r/n,/r/n同理/r/n，/r/nb/r/n4/r/n+/r/nc/r/n4/r/n±/r/n2/r/nb/r/n2/r/nc/r/n2/r/n,/r/nc/r/n4/r/n+/r/na/r/n4/r/n±/r/n2/r/na/r/n2/r/nc/r/n2/r/n,/r/n(/r/na/r/n4/r/n+/r/nb/r/n4/r/n)/r/n+/r/n(/r/nb/r/n4/r/n+/r/nc/r/n4/r/n)/r/n+/r/n(/r/nc/r/n4/r/n+/r/na/r/n4/r/n)/r/n三/r/n2/r/na/r/n2/r/nb/r/n2/r/n+/r/n2/r/nb/r/n2/r/nc/r/n2/r/n+/r/n2/r/na/r/n2/r/nc/r/n2/r/n,/r/n从而/r/na/r/n4/r/n+/r/nb/r/n4/r/n+/r/nc/r/n4/r/na/r/n2/r/nb/r/n2/r/n+/r/nb/r/n2/r/nc/r/n2/r/n+/r/nc/r/n2/r/na/r/n2/r/n./r/n1/r/n3/r/n已知/r/na/r/n>/r/nl/r/n,/r/nb/r/n>/r/n0/r/n,/r/n匸/r/n+/r/n万/r/n—/r/n1/r/n，/r/n求证：/r/na/r/n+/r/n2/r/nb/r/n±/r/n2/r/n、/r/nj6/r/n+/r/n7./r/n［/r/n证明/r/n］/r/n由/r/na/r/n+/r/nb/r/n—/r/n1/r/n，/r/n得/r/nb/r/n=/r/na/r/n_/r/n］/r/n(/r/na/r/n>/r/ni/r/n)/r/n，/r/n

/r/n6/r/na/r/n,6/r/n(/r/na/r/n—/r/n1/r/n)/r/n+/r/n6/r/n则卄/r/n2/r/n方=卄产=卄/r/na/r/n—/r/n1/r/n,6/r/nt/r/n,6,/r/n=/r/na/r/n+/r/na/r/n—/r/ni/r/n+/r/n6/r/n=/r/n(a/r/n—/r/n1)/r/n+/r/na/r/n—/r/n1/r/n+/r/n7/r/n三/r/n2/r/n卡+/r/n7/r/n,/r/na/r/n—/r/n1/r/n当/r/na/r/n—/r/n1/r/n=/r/n—/r/n1/r/n时/r/n,/r/n即/r/na/r/n=/r/n1/r/n+/r/n\/r/n；/r/n6/r/n时/r/na/r/n—/r/n1/r/na/r/n＋/r/nb/r/n1/r/n.应用基本不等式时要时刻注意其成立的条件，只有当/r/na/r/n>0/r/n,/r/nb/r/n>0/r/n时，才会/r/n有価/r/nWp/r/n対于“当且仅当/r/n时，'二'成立…”这句话要从两个方面理解：一方面，当/r/na/r/n=/r/nb/r/n时，/r/n=莎/r/n;另一方面：当仝孑=\厉/r/nb/r/n时，也有/r/na/r/n=/r/na/r/n＋/r/nb/r/n2/r/n．应用基本不等式证明不等式的关键在于进行“拼”、“凑”、“拆”、“合”、“放缩”等变形构造出符合基本不等式的条件结构/r/n../r/n当堂达标。/r/n1/r/n．/r/n思考辨析/r/nTOC\o"1-5"\h\z/r/n(1)/r/n对任意/r/na/r/n，/r/nb/r/n^/r/nR/r/n，/r/na/r/n2/r/n+/r/nb/r/n2/r/n±/r/n2/r/nab/r/n，/r/na/r/n+/r/nb/r/n±/r/n2/r/n、...//r/n0b/r/n均成立./r/n(/r/n)/r/n⑵若/r/na/r/nH/r/nO/r/n，贝/r/nU/r/na/r/n+l±/r/n2\./r/n；/r/n'a/r/n丄/r/n=/r/n2.(/r/n)/r/n(a/r/n+/r/nb/r/n⑶若/r/na/r/n>0/r/n，/r/nb/r/n>0/r/n，/r/n贝/r/nU/r/nabW/r/n—/r/n—/r/n2/r/n.(/r/n)/r/nk/r/n2/r/n丿/r/n［/r/n提示/r/n］/r/n⑴任意/r/na/r/n，/r/nb/r/n^/r/nR/r/n，/r/n有/r/na/r/n2/r/n+b/r/n2/r/n三/r/n2ab/r/n成立/r/n，/r/n当/r/na/r/n，/r/nb/r/n都为正数时/r/n，/r/n不等式/r/na/r/n+/r/nb/r/n±/r/n2/r/n、'/r/nab/r/n成立./r/n⑵/r/n只有当/r/na>0/r/n时/r/n，/r/n根据基本不等式/r/n，/r/n才有不等式/r/na+L/、/r/n丨/r/na/r/n丄=/r/n2/r/n成立/r/n./r/n(3)/r/n因为/r/n\［ab^~2~/r/n，/r/n所以/r/nabW/r/n"扌方/r/n2/r/n./r/n［答案］/r/n⑴/r/nX/r/n⑵/r/nX/r/n(3)/r/nV/r/n2/r/n.设/r/na/r/n>/r/nb/r/n>0/r/n，/r/n则下列不等式中一定成立的是/r/n()/r/na/r/nA/r/n./r/na/r/n—/r/nb/r/nvO/r/nB/r/n./r/n0</r/nb/r/n<1/r/nD/r/n./r/nab/r/n>/r/na/r/n+/r/nb/r/n

/r/nC/r/n「・・/r/na/r/n>/r/nb/r/n>0/r/n,/r/n由基本不等式知/r/nVOb/r/nv/r/n―定成立/r/n./r/n］/r/n9/r/n3/r/n．/r/n不等式/r/n=+/r/n（/r/nx/r/n—/r/n2）/r/n2/r/n6（/r/n其中/r/nx/r/n>/r/n2/r/n）/r/n中等号成立的条件是/r/n3/r/n．/r/nA/r/n./r/nx/r/n=/r/n3/r/nB/r/n./r/nx/r/n=—/r/n3/r/nC/r/n./r/nx/r/n=/r/n5/r/nD/r/n./r/nx/r/n=_/r/n5/r/n［/r/n由基本不等式知等号成立的条件为/r/n9/r/nx/r/n2/r/n=/r/nx/r/n—/r/n2/r/n,/r/n即/r/nx/r/n=/r/n5（/r/nx/r/n=/r/n—/r/n1/r/n舍去/r/n）/r/n./r/n]/r/n4/r/n./r/nb/r/n2/r/na/r/n2/r/n设/r/na/r/n>0/r/n,/r/nb/r/n>0/r/n,/r/n证明：万+万/r/n2/r/na/r/n+/r/nb/r/n-/r/n[/r/n证明/r/n]/r/n・/r/na/r/n>0/r/n,/r/nb/r/n>0/r/n,/r/n•普+於/r/n2/r/n方’労/r/n+/r/nb/r/n2/r/n2a/r/n，/r/n••竺/r/n+¥±/r/na/r/n+/r/nb/r/n./r/nab/r/n分层作业基本不等式/r/n（/r/n建议用时：/r/n60/r/n分钟/r/n）/r/n［/r/n合格基础练/r/n］/r/n一/r/n、/r/n选择题/r/n1/r/n.设/r/nt/r/n=/r/na/r/n+/r/n2/r/nb/r/n,/r/ns/r/n=/r/na/r/n+/r/nb/r/n2/r/n+/r/nl/r/n,/r/n则/r/nt/r/n与/r/ns/r/n的大小关系是/r/n（/r/n）/r/nA/r/n./r/ns/r/n2/r/nt/r/nB/r/n./r/ns/r/n>/r/nt/r/nC/r/n./r/nsWt/r/nD/r/n./r/ns/r/n</r/nt/r/nA/r/n[•••b/r/n2/r/n+/r/n1/r/n±/r/n2/r/nb/r/n,・・・/r/na/r/n+/r/n2/r/nb/r/nW/r/na/r/n+/r/nb/r/n2/r/n+/r/n1./r/n]/r/n2/r/n.下列不等式中正确的是/r/n（/r/n）/r/nA/r/n./r/na/r/n+/r/n4/r/n2/r/n4/r/na/r/nB/r/n./r/na/r/n2/r/n+/r/nb/r/n2/r/n2/r/n4/r/nab/r/n4/r/nD/r/n［/r/naV/r/nO/r/n,/r/n则/r/na/r/n+-/r/n2/r/n4/r/n不成立/r/n，/r/n故/r/nA/r/n错;/r/n

/r/na/r/na/r/n=/r/n1/r/n,/r/nb/r/n=/r/n1/r/n,/r/na/r/n2/r/n+/r/nb/r/n2/r/nV/r/n4/r/nab/r/n,/r/n故/r/nB/r/n错;/r/na=/r/n4/r/n，/r/nb=/r/n16/r/n，/r/n则/r/n\/r/n；/r/nGbV/r/n竺$，故/r/nC/r/n错;/r/n由基本不等式可知/r/nD/r/n项正确/r/n．/r/n］/r/n3/r/n.已知/r/na/r/n>0/r/n,/r/nb/r/n>0/r/n,/r/n则下列不等式中错误的是/r/n（/r/n）/r/nA/r/n./r/nab/r/nW/r/n(/r/na/r/nA/r/n./r/nab/r/nW/r/n(/r/na/r/n+/r/nb/r/n\/r/n2/r/na/r/n2/r/n+/r/nb/r/n2/r/nB/r/n./r/nab/r/nW/r/n—/r/n2/r/n—/r/nC/r/n丄三「/r/naba/r/n2/r/n+/r/nb/r/n2/r/n(/r/n2/r/n)/r/n2/r/nabf/r/n(/r/na/r/n+/r/nb/r/n丿/r/nD.A/r/nW/r/n(/r/na/r/n+/r/nb/r/n\/r/n2/r/n(/r/na/r/n+/r/nb/r/n\/r/n2/r/n，・・/r/nab=(a/r/n+/r/nb/r/n丿/r/n［/r/n由基本不等式知/r/nA/r/n、/r/nC/r/n正确，由重要不等式知/r/nB/r/n正确，由/r/n三/r/nab/r/n，・・/r/nab=(a/r/n+/r/nb/r/n丿/r/n故选/r/nD.］/r/n4/r/n./r/n4/r/n./r/n若/r/na/r/n>/r/nb/r/n>/r/n0/r/n,/r/n则下列不等式成立的是/r/n（）/r/nA/r/nA/r/n./r/nt/r/na/r/nI/r/nb/r/nt/r/n—/r/na/r/n>/r/nb/r/n>/r/n~2/r/n~>\/r/n'ab/r/nB/r/n./r/na/r/n+/r/nb/r/na/r/n>一/r/n2~/r/n>\/r/nab/r/n>/r/nb/r/nC/r/n./r/na/r/n+/r/nb/r/na/r/n>/r/n—/r/n2~>b>-jab/r/na/r/nI/r/nb/r/nD./r/na/r/n〉\：/r/nab/r/n>/r/n?/r/n>/r/nb/r/na+aa+b/r/nt/r/n—/r/n,/r/n——/r/n—/r/n，/r/n亠/r/n“/r/nB/r/n[a^—^/r/n-/r/n^~2~/r/n>\；/r/nab/r/n〉/r/nJb/r/n・/r/nb/r/n=/r/nb/r/n，/r/n因此只有/r/nB/r/n项正确/r/n./r/n]/r/n5/r/n.若/r/na/r/n>/r/n0/r/n,/r/nb/r/n>/r/n0/r/n,/r/n且/r/na/r/n+/r/nb/r/n=/r/n4/r/n，则下列不等式恒成立的是/r/n（/r/n）/r/n11/r/nA./r/nab/r/n>/r/n2/r/nB./r/n1/r/n+/r/nb/r/nW/r/n1/r/nab/r/nC./r/n、/r/n拓/r/nb/r/n±/r/n2/r/n1/r/nW/r/n1/r/nD°a/r/n2/r/n+/r/nb/r/n2f/r/n8/r/nD/r/n[/r/n由/r/n\/r/n：/r/nab/r/nW/r/n2/r/n得/r/nab/r/nW/r/n4/r/n,/r/n・•・/r/n加/r/n4/r/n,故/r/nA/r/n错;/r/n.1.1/r/na/r/n+/r/nb/r/n4/r/n、/r/n

/r/nB/r/n中/r/n，/r/na/r/n+/r/nb/r/n=/r/n"ar/r/n=/r/nab/r/n^/r/nL/r/n故/r/nB/r/n错；/r/n由/r/na/r/n+/r/nb/r/n=/r/n4/r/n，/r/n得/r/n\/r/n；/r/n'ab/r/nW/r/n空尹/r/n=/r/n2/r/n=/r/n2/r/n，/r/n故/r/nC/r/n错;/r/na/r/n2/r/n+b2/r/n(/r/na+b/r/n\/r/n2/r/n得/r/na/r/n2/r/n+/r/nb/r/n2/r/n±/r/n2/r/nX/r/n(/r/n2/r/n丿/r/n2/r/n=/r/n8,/r/n・・・/r/nark/r/n誌/r/nD/r/n正确/r/n•/r/n]/r/n二/r/n、/r/n填空题/r/na/r/n—/r/nc/r/n6/r/n.已知/r/na/r/n〉/r/nb/r/n〉/r/nc/r/n,/r/n贝灯/r/n(/r/na/r/n—/r/nb/r/n)(/r/nb/r/n—/r/nc/r/n丿/r/n与/r/n一/r/n^/r/n-/r/n的大小关系是/r/n\；(/r/na——b/r/n)(/r/nb——c/r/n)/r/nW/r/n2/r/n「・°/r/na/r/n>/r/nb/r/n>/r/nc,/r/n(a——b)/r/n+/r/n(b——c)/r/na——c/r/n.•・\；(/r/na/r/n——/r/nb/r/n)(/r/nb/r/n——/r/nc/r/n)/r/nW/r/n2/r/n=^^.]/r/n7/r/n.某工厂第一年的产量为/r/nA/r/n,/r/n第二年的增长率为/r/na/r/n,/r/n第三年的增长率为/r/nb/r/n则这两年的平均增长率/r/nx/r/n与增长率的平均值孚的大小关系为./r/nxW/r/n—/r/n2—/r/n[/r/n用两种方法求出第三年的产量分别为/r/nA/r/n(l/r/n+/r/na/r/n)(l/r/n+/r/nb/r/n)/r/n,/r/nA/r/n(l/r/n+/r/nx/r/n)/r/n2/r/n,/r/n则有/r/n(1/r/n+/r/nx/r/n)/r/n2/r/n=/r/n(l/r/n+/r/na/r/n)(l/r/n+/r/nb/r/n)/r/n./r/n・・・/r/nl+x=/r/nM+/r/na/r/n)(/r/nl+/r/nb/r/n)/r/nW/r/nl/r/n+/r/na/r/n+/r/nl/r/n+/r/nb/r/n=l/r/n+/r/na/r/n+/r/nb/r/n・・/r/nxW"/r/n扌方.当且仅当/r/na=b/r/n时等号成立/r/n./r/n]/r/n8/r/n.已知函数/r/nfx/r/n)/r/n=/r/n4/r/nx/r/n+/r/na/r/n(/r/nx/r/n>0/r/n,/r/na/r/n>0/r/n)/r/n在/r/nx/r/n=/r/n3/r/n时取得最小值，则/r/na/r/n=/r/n当且仅当/r/n4/r/n当且仅当/r/n4/r/nx/r/n=/r/na/r/n,/r/n即/r/nx/r/n=/r/n2/r/na/r/n时等号成立/r/n，/r/n此时/r/nfx/r/n)/r/n取得/r/nx/r/n2/r/n36/r/n[/r/nfx/r/n)/r/n=/r/n4/r/nx+~/r/n三/r/n2\!/r/n4/r/nx/r/n・/r/n-/r/n=4\/r/n：/r/na/r/n(/r/nx/r/n>0/r/n，/r/na/r/n>0),/r/nx\Jx/r/n最小值/r/n4&./r/n又由已知/r/nx/r/n=/r/n3/r/n时/r/n，/r/nfx/r/n)/r/nmin/r/n=/r/n4/r/n寸/r/na/r/n，/r/n•••¥=/r/n3/r/n,/r/n即/r/na/r/n=/r/n36.]/r/n三/r/n、/r/n解答题/r/n9/r/n.已知/r/n9/r/n.已知/r/na/r/n，/r/nb/r/n，/r/nc/r/n为正实数，且/r/na/r/n+/r/nb/r/n=/r/nl./r/n求证:/r/nl/r/n+/r/nl/r/n^/r/n4./r/nab/r/nl,l/r/na/r/n+/r/nb/r/ni/r/na/r/n+/r/nb/r/n[/r/n证明/r/n]/r/na/r/n+/r/nb/r/n=〒+—/r/nr/r/n=/r/nl/r/n+/r/nb/r/n+/r/nb/r/n+/r/nl/r/nab/r/n=/r/n2/r/n+/r/nb/r/n+/r/n診/r/n2/r/n+/r/n2/r/n」/r/nb/r/n・/r/nb/r/n=/r/n4/r/nab/r/n\ab/r/n当且仅当/r/na/r/n=/r/nb/r/n时/r/n“/r/n=/r/n”/r/n成立/r/n./r/n

/r/nb/r/n＋/r/nc/r/n－/r/nac/r/n＋/r/na/r/n－/r/nba/r/n＋/r/nb/r/n－/r/nc/r/nTOC\o"1-5"\h\z/r/n10/r/n.已知/r/na/r/n、/r/nb/r/n、/r/nc/r/n为正数，求证：/r/nH/r/nb/r/nH/r/n±/r/n3./r/na/r/nb/r/nc/r/nb.c/r/nc.a/r/na.b_/r/n［/r/n证明/r/n］/r/n左边/r/n=-/r/n+-一/r/n1/r/n+：+—/r/n1/r/n+-+-一/r/n1/r/naa/r/nbb/r/ncc/r/n=/r/n(/r/na/r/n+/r/nb/r/n)/r/n+/r/n(/r/na/r/n+/r/na/r/n)/r/n+/r/n(/r/nc/r/n+/r/nb/r/n］/r/n-/r/n3./r/nV/r/na/r/n,/r/nb/r/n,/r/nc/r/n为正数/r/n,/r/nba/r/n・・・/r/na/r/n+/r/nb/r/n±/r/n2/r/n(/r/n当且仅当/r/na/r/n=/r/nb/r/n时取/r/n“/r/n=/r/n”/r/n)/r/n；/r/nca/r/n-/r/n+/r/na/r/n±/r/n2(/r/n当且仅当/r/na/r/n=/r/nc/r/n时取/r/n“/r/n=/r/n”/r/n)/r/n;/r/nac/r/ncb/r/nb/r/n+/r/nc/r/n±/r/n2（/r/n当且仅当/r/nb=c/r/n时取/r/n“/r/n=/r/n”/r/n）/r/n./r/n从而/r/n*6（/r/n当且仅当/r/na/r/n=/r/nb/r/n=/r/nc/r/n时取等号）./r/nb/r/n+/r/nb/r/n)/r/n+/r/n(/r/na/r/n+/r/na/r/n)/r/n+/r/n(/r/nb/r/n+/r/nl/r/n)/r/n-/r/n3/r/n±/r/n3,/r/nb/r/n+/r/nc/r/n－/r/nac/r/n+/r/na/r/n－/r/nba/r/n+/r/nb/r/n－/r/nc/r/n即/r/n—/r/n+^—+^—/r/n沦/r/n[/r/n等级过关练/r/n]/r/n1/r/n．下列不等式一定成立的是/r/n（/r/nA/r/n./r/nx/r/n+/r/nx/r/n±/r/n2/r/nb.^/r/n+^汽/r/nA/r/n./r/nx/r/n+/r/nx/r/n±/r/n2/r/nD/r/n./r/n2/r/n-/r/n3/r/nD/r/n./r/n2/r/n-/r/n3/r/nx/r/n-/r/n4/r/n±/r/n2/r/nx/r/nC.R/r/n2/r/nB/r/n[A/r/nB/r/n[A/r/n项中当/r/nx/r/n<0/r/n时/r/n，/r/nx/r/n+丄/r/n<0v2,/r/nx/r/n・/r/nA/r/n错误/r/n．/r/nx/r/n2/r/n+/r/n2/r/n・/r/nB/r/n正确/r/n．/r/nB/r/n项中，袞/r/n+r/r/n、/r/nx/r/n2/r/n+/r/n2/r/n・/r/nB/r/n正确/r/n．/r/nx/r/n2/r/n+/r/n3/r/n3/r/n当/r/nx/r/n=/r/n0/r/n时/r/n，/r/n卞百=/r/n2<2/r/n,/r/n显然选项/r/nC/r/n不正确./r/n4/r/nD/r/n项中取/r/nx/r/n=/r/n1,2/r/n-/r/n3/r/nx/r/n—/r/n一/r/nv2/r/n,・・・/r/nD/r/n错误/r/n./r/n]/r/nx/r/n2/r/n.已知/r/na/r/n±/r/n0/r/n,/r/nb/r/n±/r/n0/r/n,/r/n且/r/na/r/n+/r/nb/r/n=/r/n2/r/n,/r/n贝肚/r/n)/r/n

/r/nB/r/n./r/nab/r/n三/r/n2/r/nA/r/nB/r/n./r/nab/r/n三/r/n2/r/nC/r/n./r/na/r/n2/r/n+/r/nb/r/n2/r/n±/r/n2/r/nD/r/n./r/na/r/n2/r/n+/r/nb/r/n2/r/nW/r/n3/r/n(a+/r/nb/r/n)/r/n2/r/nC/r/n[/r/nT/r/na/r/n^/r/nO/r/n,/r/nb/r/n三/r/n0/r/n,/r/n且/r/na+b=/r/n2/r/n,：・/r/nabW—^/r/n=/r/n1/r/n,/r/nk/r/n(/r/na/r/n+/r/n(/r/na/r/n+/r/nb/r/n)/r/n2/r/n(/r/na/r/n+/r/nb/r/n\/r/na/r/n+/r/nb/r/nab/r/n=—/r/n=M/r/n2/r/n'/r/n故/r/n②/r/n正确；对于/r/n③，/r/n当/r/na/r/n=/r/nb/r/n=T/r/n时/r/n，/r/n不等式的左边为丁=-/r/n1/r/n,/r/n右边为均=/r/n可知/r/n③/r/n不正确；令/r/na/r/n=/r/n1/r/n,/r/nb/r/n=-/r/n1/r/n可知/r/n④/r/n不正确/r/n./r/n]/r/n5/r/n.已知/r/na/r/n、/r/nb/r/n、/r/nc/r/n为不全相等的正实数，求证：/r/na/r/n+/r/nb/r/n+/r/nc>"Jab+“Jbc+“Jca/r/n./r/n[/r/n证明/r/n]/r/nV/r/na>/r/n0/r/n,/r/nb>/r/n0/r/n,/r/nc>/r/n0/r/n./r/n.a/r/n+/r/nb/r/nb/r/n+/r/nc/r/nc/r/n+/r/na/r/na/r/n+/r/nbb/r/n+/r/ncc/r/n+/r/na/r/n・°・/r/n一三*/r/nab/r/n,/r/n2/r/n三/r/n'Jbc/r/n,/r/n2/r/n'ca/r/n,.°・/r/n一+/r/n+/r/n2/r/n'ab/r/n+\'/r/n'bc/r/n+/r/n-Jca/r/n,/r/n即/r/na/r/n+/r/nb/r/n+/r/ncM.yOb/r/n+/r/n瓦/r/n+/r/n\i'ca./r/n由于/r/na/r/n、/r/nb/r/n、/r/nc/r/n不全相等,/r/n・/r/n等号不成立/r/n，/r/n.*.a+b+c＞/r/n；/r/n'ab/r/n+\；/r/nbc+-•Jca/r/n./r/n而/r/n4/r/n=/r/n(/r/na/r/n+/r/nb/r/n)/r/n2/r/n=/r/na/r/n2/r/n+/r/nb/r/n2/r/n+/r/n2/r/nab/r/nW/r/n2(/r/na/r/n2/r/n+/r/nb/r/n2/r/n)/r/n,/r/na/r/n2/r/n+/r/nb/r/n2/r/n±/r/n2.]/r/n3/r/n.若/r/nx/r/n2/r/n+/r/ny/r/n2/r/n=/r/n4/r/n，则/r/nxy/r/n的最大值为/r/n./r/nx/r/n2/r/n+/r/ny/r/n2/r/n2/r/n[/r/nxy/r/nW2/r/n—/r/n=/r/n2/r/n,/r/n当且仅当/r/nx/r/n=/r/ny/r/n时取/r/n“/r/n=/r/n"./r/n]/r/n4/r/n.设/r/na/r/n,/r/nb/r/n为非零实数，给出不等式：/r/na/r/n2/r/n+/r/nb/r/n2/r/n2/r/n三/r/nab/r/na/r/n2/r/n+/r/nb/r/n2/r/n2/r/n三/r/nab/r/n；/r/na/r/n2/r/n+/r/nb/r/n2/r/n2/r/n(/r/na/r/n+/r/nb/r/n\/r/na/r/n+/r/nb/r/n③丁/r/nab/r/n=a+b'/r/na/r/n④沪/r/n±/r/n2./r/n其中恒成立的不等式是/r/n①②/r/n[/r/n由重要不等式/r/na/r/n2/r/n+/r/nb/r/n2/r/n±/r/n2/r/nab/r/n可知/r/n①/r/n正确;/r/na/r/n2/r/n+/r/nb/r/n2/r/n2/r/n(/r/na/r/n2/r/n+/r/nb/r/n2/r/n)/r/n②/r/n4/r/n(/r/na/r/n2/r/n+/r/nb/r/n2/r/n)/r/n+/r/n(/r/na/r/n2/r/n+/r/nb/r/n2/r/n)/r/na/r/n2/r/n+/r/nb/r/n2/r/n+/r/n2/r/nab/r/n

/r/n第/r/n2/r/n课时^基本不等式的应用/r/n自主预习。探新/r/n堀/r/ni/r/n：/r/n£iiHUYUXlTAINXIM^HL/r/n匚/r/n新和初探/r/n已知/r/nx/r/n、/r/ny/r/n都是正数，/r/nS/r/n2/r/n⑴若/r/nx/r/n+/r/ny/r/n=/r/nS/r/n（/r/n和为定值/r/n）/r/n，则当/r/nx/r/n=/r/ny/r/n时，积/r/nxy/r/n取得最大值/r/n才./r/n（2）/r/n若/r/nxy/r/n=/r/np/r/n（/r/n积为定值/r/n）/r/n，则当/r/nx/r/n=/r/ny/r/n时，和/r/nx/r/n+/r/ny/r/n取得最小值/r/n2\®./r/n上述命题可归纳为口诀：积定和最小，和定积最大./r/n」初试身手/r/n14/r/n1/r/n.已知/r/na/r/n>/r/n0/r/n,/r/nb/r/n〉/r/n0/r/n,/r/na/r/n+/r/nb/r/n=/r/n2/r/n,/r/n则/r/ny/r/n=/r/n^/r/n+/r/nb/r/n的最小值是/r/n（/r/n）/r/nA./r/n7/r/nA./r/n7/r/nB/r/n./r/n4/r/nC./r/n9/r/nD/r/n./r/n5/r/na/r/n＋/r/nb/r/nC/r/n「••/r/na/r/n+/r/nb/r/n=/r/n2/r/n,・・・一^=/r/n1./r/n・/r/n1/r/n斗/r/n4/r/n_/r/n（/r/n1/r/n丰/r/n4/r/n）（/r/na/r/n+/r/nb'/r/n…/r/na/r/nb=\a/r/nb/r/nj]/r/n2/r/n丿/r/nTOC\o"1-5"\h\z/r/n5,/r/n（/r/n2a|/r/nb/r/n5/r/nlOb/r/n9/r/n=/r/n2/r/n+/r/nIt/r/n+/r/n2/r/na/r/n戶/r/n2/r/n+2/r/n\/r/n；/r/n万无=/r/n2/r/n[/r/n当且仅当/r/n¥=寻，/r/n即/r/nb=/r/n2/r/na/r/n时/r/n，/r/n等号成立）/r/n1/r/n4/r/n9/r/n故/r/ny/r/n=/r/na/r/n+/r/nb/r/n的最小值为/r/n2/r/n」/r/n2/r/n2/r/n.若/r/nx/r/n>0/r/n，/r/n则/r/nx/r/n+/r/n-/r/n的最小值是/r/n./r/nx/r/n2/r/n//r/n~2/r/n2/r/n、/r/n2/r/n[/r/nx/r/n+■—/r/n三/r/n2/r/n飞;/r/n'x/r/n・-/r/n=/r/n2\/2/r/n，/r/n当且仅当/r/nx/r/n=j/r/n2/r/n时/r/n，/r/n等号成立/r/n./r/n]/r/nx/r/n\t/r/nx/r/n设/r/nx/r/n，/r/ny/r/n^/r/nN/r/n*/r/n满足/r/nx/r/n+/r/ny/r/n=/r/n20/r/n，则/r/nxy/r/n的最大值为/r/n./r/n100/r/n[/r/nV/r/nx/r/n，/r/ny/r/nG/r/nN/r/n*，・°・/r/n20/r/n=/r/nx/r/n+/r/ny/r/n±2\；/r/nxy/r/n，/r/n・/r/n°/r/n・/r/nxy/r/nW/r/n100.]/r/n合作探究毎素养/r/nEl£2UOrANJIU/r/nJ'L/r/nSUYAN/r/n{:/r/n

/r/n利用基本不等式求最值/r/n兴空/r/n1//r/n【例/r/n1/r/n】/r/n⑴/r/n已知/r/nx/r/nv#/r/n,/r/n求/r/ny/r/n=/r/n4/r/nx/r/n-/r/n2/r/n+/r/n4/r/n丄/r/n5/r/n的最大值；/r/n(2)/r/n已知/r/n0</r/nx/r/n<2/r/n，/r/n求/r/ny/r/n=/r/n*/r/nx/r/n(1/r/n—/r/n2/r/nx/r/n)/r/n的最大值./r/n［/r/n思路点拨/r/n］/r/n(1)/r/n看到求/r/ny/r/n=/r/n4/r/nx/r/n-/r/n2/r/n+/r/n4/r/nX/r/n^/r/n5/r/n的最值/r/n，/r/n想到如何才能出现乘积定值；/r/n(/r/n2)/r/n要求/r/ny/r/n=/r/n|/r/nx/r/n(1/r/n-/r/n2/r/nx/r/n)/r/n的最值/r/n,/r/n需要出现和为定值/r/n./r/n［/r/n解/r/n］/r/n(1)/r/nT/r/nx/r/n<4/r/n'/r/n・・・/r/n5/r/n-/r/n4/r/nx/r/n>0/r/n,/r/n1/r/n(/r/n,/r/n1/r/nA/r/n・/r/ny/r/n=/r/n4x/r/n_/r/n2/r/n+/r/n4/r/nX/r/n^/r/n5/r/n=_/r/nl/r/n5/r/n_/r/n4x/r/n+/r/n5/r/n^/r/n4/r/nX/r/nj/r/n+/r/n3/r/n<_/r/n2/r/n+/r/n3/r/n=/r/n1/r/n，/r/n当且仅当/r/n5/r/n—/r/n4/r/nx/r/n=/r/n5/r/n^/r/n4/r/nX/r/n，/r/n即/r/nx/r/n—/r/n1/r/n时/r/n，/r/n上式等号成立/r/n，/r/n故当/r/nx/r/n—/r/n1/r/n时/r/n，/r/ny/r/nmax/r/n—/r/n1/r/n-/r/n⑵/r/n0v/r/nx/r/n<2/r/n，/r/n・/r/n1/r/n－/r/n2/r/nx/r/n>0/r/n，/r/n./r/n1/r/n—1/r/n(/r/n2/r/nx/r/n+/r/n1/r/n—/r/n2/r/nx/r/n)/r/n111/r/n-/r/n-/r/ny/r/n=/r/n4/r/nX/r/n2x(1/r/n_/r/n2x)/r/n^/r/n4/r/nX/r/nl^/r/n2/r/nJ/r/n2/r/n=/r/n4/r/nX/r/n4/r/n=/r/n16./r/n・/r/n••当且仅当/r/n2/r/nx/r/n=/r/n1/r/n—/r/n2/r/nx/r/nf/r/n0v/r/nx/r/nv|/r/nj/r/n,/r/n即/r/nx/r/n=/r/n4/r/n时/r/n，/r/ny/r/nmax/r/n=/r/n16/r/n-/r/n观律方法/r/n利用基本不等式求最值的关键是获得满足基本不等式成立条件/r/n，/r/n即/r/n“/r/n一正/r/n、/r/n二定/r/n、/r/n三相等/r/n”/r/n./r/n解题时应对照已知和欲求的式子运用适当的/r/n“/r/n拆项/r/n、/r/n添项/r/n、/r/n配凑/r/n、/r/n变形/r/n”/r/n等方法创设应用基本不等式的条件/r/n./r/n具体可归纳为三句话：若不正/r/n，/r/n用其相反数/r/n，/r/n改变不等号方向；若不定应凑出定和或定积；若不等/r/n，/r/n一般用后面第三章/r/n§3.2/r/n函数的基本性质中学习/r/n./r/n跟踪训练/r/n1/r/n./r/n(1)/r/n已知/r/n跟踪训练/r/n1/r/n./r/n(1)/r/n已知/r/nx/r/n>0/r/n，/r/n求函数/r/ny/r/n=/r/nx/r/n2/r/n+/r/n5/r/nx/r/n+/r/n4/r/nx/r/n的最小值;/r/n(2)/r/n已知/r/nOv/r/nx/r/nvg/r/n，/r/n求函数/r/ny/r/n=/r/nx/r/n(1/r/n—/r/n3/r/nx/r/n)/r/n的最大值./r/nx/r/n2/r/n+/r/n5/r/nx/r/n+/r/n4/r/n,4,/r/n［/r/n解/r/n］/r/n(1)/r/nT/r/ny/r/n=/r/n-—/r/n=/r/nx/r/n+/r/nX/r/n+/r/n5/r/n±/r/n2\/r/n：/r/n4/r/n+/r/n5/r/n=/r/n9/r/n,/r/nx/r/nx/r/n4/r/n当且仅当/r/nx/r/n=/r/nx/r/n即/r/nx/r/n=/r/n2/r/n时等号成立./r/n故/r/ny/r/n=/r/n故/r/ny/r/n=/r/nx/r/n2/r/n+/r/n5/r/nx/r/n+/r/n4/r/nx/r/n(/r/nx/r/n>0)/r/n的最小值为/r/n9./r/n法一/r/n：/r/n法一/r/n：/r/n*.*0</r/nx/r/n<3/r/n,A/r/n1/r/n—/r/n3/r/nx/r/n>0./r/n・/r/n°/r/n・/r/ny/r/n=/r/nx/r/n(l/r/n—/r/n3/r/nx/r/n)/r/n=/r/n3,3/r/nx/r/n(1/r/n—/r/n3/r/nx/r/n)/r/n”13/r/nx+/r/n(/r/n1/r/n—/r/n3/r/nx/r/n)］/r/n2/r/n_/r/n1/r/n汽/r/n2/r/nJ/r/n=/r/n12/r/n./r/n当且仅当/r/n3/r/nx/r/n=/r/n1/r/n—/r/n3/r/nx/r/n,/r/n即/r/nx/r/n=/r/n6/r/n时/r/n，/r/n等号成立./r/n当/r/nx/r/n=/r/n6/r/n时/r/n，/r/n函数取得最大值吉./r/n法二/r/n：/r/n*.*0<x<3,/r/n法二/r/n：/r/n*.*0<x<3,/r/n.*.3—x>0./r/n(/r/n(/r/n1/r/n)/r/n・・/r/ny/r/n=/r/nx/r/n(1/r/n—/r/n3/r/nx/r/n)/r/n=/r/n3.x£/r/n—/r/nx/r/nj/r/nw/r/n3/r/n・/r/nX/r/n+/r/n3X/r/n2/r/nk/r/n2/r/n丿/r/n_丄/r/n=/r/n12/r/n当且仅当/r/nx/r/n=/r/n当且仅当/r/nx/r/n=/r/n3/r/n—/r/nx/r/n,/r/n即/r/nx/r/n=/r/n6/r/n时/r/n，/r/n等号成立.当/r/nx/r/n=/r/n6/r/n时/r/n，/r/n函数取得最大值吉./r/n利用基本不等式求条件最值/r/n例/r/n2/r/n例/r/n2/r/n】/r/n81/r/n已知/r/nx/r/n>/r/n0/r/n,/r/ny/r/n>/r/n0/r/n,/r/n且满足/r/n兀/r/n+/r/n歹/r/n一/r/n1./r/n求/r/nx/r/n+/r/n2/r/ny/r/n的取小值./r/n81/r/n［/r/n解/r/n］/r/n・・・/r/nx/r/n>/r/n0/r/n,/r/ny/r/n>/r/n0/r/n,/r/nx/r/n+/r/ny/r/n=/r/n1/r/n，-/r/nx/r/n+/r/n2y/r/n=/r/n(/r/n!/r/n+/r/ny/r/n)/r/n(x/r/n+/r/n2y)/r/n=/r/n10/r/n+/r/ny/r/n+/r/n呼日/r/n0/r/n+/r/n2\/r/n罟/r/n18/r/n,/r/n当且仅当/r/n'/r/n当且仅当/r/n'/r/n8/r/n+】=/r/n1,/r/nxy/r/nx_/r/n16/r/ny/r/n—/r/n,/r/n即/r/n［/r/nx/r/n—/r/n;/r/n2/r/n时/r/n，/r/n等号成立,/r/nl/r/ny/r/n=/r/n3/r/n故当/r/ni/r/n=/r/n12/r/n,/r/ny/r/n=/r/n3/r/n时/r/n，/r/n（/r/ni/r/n+/r/n2/r/ny/r/n）/r/nmin/r/n=/r/n18./r/n［母题探究］/r/n81/r/n81/r/n若把/r/n“8/r/n+/r/ny/r/n=/r/n1/r/n”/r/n改为/r/n“/r/ni/r/n+/r/n2/r/ny/r/n=/r/n1/r/n”/r/n，其他条件不变，求/r/n；/r/n+/r/n-/r/n的最小值./r/n若把/r/niy/r/n卜/r/nb）/r/n卜/r/nb）/r/n（/r/na/r/n+/r/n2b/r/n）/r/n［/r/n解/r/n［/r/n解/r/n］/r/nV/r/nx/r/n,/r/ny/r/n^/r/nR/r/n丰/r/n=/r/n（/r/nx/r/n+/r/n2y/r/n）/r/n〔/r/ni+y/r/n=/r/n8/r/n+呼+/r/n：/r/n+/r/n2/r/n=/r/n10/r/n+竿+詩/r/n10/r/n+/r/n2/r/n帀/r/n=/r/n18/r/n当且仅当竿=;时取等号,/r/n21/r/n结/r/ni/r/n+/r/n2/r/ny/r/n—/r/n1/r/n,/r/n得/r/ni/r/n=/r/n3/r/n，/r/n尹=/r/n6/r/n'/r/n2181/r/n・/r/n••当/r/ni/r/n=/r/n3/r/n，/r/ny/r/n=/r/n6/r/n时/r/n，/r/ni/r/n+/r/ny/r/n取到最小值/r/n18./r/n1/r/n．/r/n本题给出的方法/r/n，/r/n用到了基本不等式/r/n，/r/n并且对式子进行了变形/r/n，/r/n配凑出满足基本不等式的条件/r/n，/r/n这是经常使用的方法/r/n，/r/n要学会观察/r/n、/r/n学会变形/r/n．/r/nb/r/n常见的变形技巧有：/r/n（/r/n1）/r/n配凑系数；/r/n（/r/n2）/r/n变符号；/r/n（/r/n3）/r/n拆补项/r/n./r/n常见形式有/r/nfx/r/n）/r/n=/r/nax/r/n+/r/nx/r/n型和/r/nf/r/n（/r/nx/r/n）/r/n=ax/r/n（/r/nb/r/nx/r/n—/r/nai/r/n）/r/n型/r/n./r/n㊃/r/n跟踪观练/r/n2/r/n.已知/r/na/r/n>/r/n0/r/n，/r/nb/r/n〉/r/n0/r/n,/r/na/r/n+/r/n2/r/nb/r/n=/r/n1/r/n，/r/n求的最小值・/r/n［/r/n解/r/n］/r/n［/r/n解/r/n］/r/n法/r/n1/r/n+/r/n1/r/nab/r/nD/r/n1/r/n=/r/n1/r/n+¥+a+/r/n2/r/n=/r/n3/r/n+/r/n2b/r/n‘/r/na/r/n卜訐/r/n3/r/n+/r/n2/r/n2/r/nb/r/na/r/nab/r/n=/r/n3/r/n+/r/n2*2/r/n，/r/n'/r/n2/r/nb/r/n—/r/na/r/nJ/r/na=^/2_1,/r/n时等号成立/r/n./r/n当且仅当/r/n＜/r/na/r/nb/r/n，/r/n即/r/n＜/r/n寸/r/n时等号成立/r/n./r/n、/r/na+/r/n2/r/nb=/r/n1/r/n,/r/nl/r/nb/r/n=/r/n】/r/n—/r/n2/r/n・・・/r/n1/r/n+/r/n1/r/n的最小值为/r/n3/r/n+/r/n2/r/n羽./r/n法二/r/n：/r/n法二/r/n：/r/na/r/n+/r/nb/r/n=〒/r/n+/r/nb/r/n=/r/n1/r/n+/r/n2b/r/n+/r/nb/r/n+/r/n2/r/nb/r/nab/r/n=/r/n3/r/n+乎+詩/r/n3/r/n+/r/n=/r/n3/r/n+乎+詩/r/n3/r/n+/r/n2/r/n佰/r/nr/r/n2b/r/n=/r/na/r/n当且仅当/r/n</r/na/r/nb/r/n'/r/n、/r/na/r/n+/r/n2/r/nb/r/n=/r/n1/r/n,/r/nA/r/n1/r/n+/r/n1/r/n的最小值为/r/n3/r/n+/r/n2/r/n叮/r/n2./r/n利用基本不等式解决实际问题/r/n时/r/n，/r/n等号成立/r/n，/r/n空型/r/n3//r/n【例/r/n3/r/n】如图，动物园要围成相同面积的长方形虎笼四间，一面可利用原有的墙，其他各面用钢筋/r/n【例/r/n3/r/n】如图，动物园要围成相同面积的长方形虎笼四间，一面可利用原有的墙，其他各面用钢筋/r/n网围成．现有/r/n36/r/nm/r/n长的钢筋网材料，每间虎笼的长、宽分别设计为多少时，可使每间虎笼面积最大？/r/n［/r/n解/r/n］/r/n设每间虎笼长/r/nx/r/nm/r/n,/r/n宽/r/ny/r/nm/r/n,/r/n则由条件知/r/n，/r/n4/r/nx/r/n+/r/n6/r/ny/r/n=/r/n36/r/n,/r/n即/r/n2/r/nx/r/n+/r/n3/r/ny/r/n=/r/n18./r/n设每间虎笼面积为/r/nS/r/n,/r/n则/r/nS/r/n=/r/nxy/r/n./r/n法一：/r/n由于/r/n2/r/nx/r/n+/r/n3/r/ny/r/n±/r/n2\/r/n：/r/n'2/r/nx/r/n・/r/n3/r/ny/r/n=/r/n2\/6/r/nXy/r/n,/r/n所以/r/n2/r/n、/r/nj6/r/nxy/r/nW/r/n18/r/n,/r/n得/r/nxyW#/r/n,/r/n即/r/nS/r/nmax/r/n即/r/nS/r/nmax/r/n=/r/n27/r/n

/r/n亍/r/n当且仅当/r/n2/r/nx/r/n=/r/n3/r/ny/r/n时/r/n，/r/n等号成立/r/n．/r/n2/r/nx/r/n+/r/n3/r/ny/r/n=/r/n18/r/n，/r/nx/r/n=/r/n4.5/r/n，/r/n由/r/n］/r/n解得/r/n］/r/n〔/r/n2/r/nx/r/n=/r/n3/r/ny/r/n,/r/n〔/r/ny/r/n=/r/n3./r/n故每间虎笼长为/r/n4.5m/r/n,/r/n宽为/r/n3/r/nm/r/n时/r/n，/r/n可使每间虎笼面积最大./r/n3/r/n法—/r/n：/r/n由/r/n2/r/nx/r/n+/r/n3/r/ny/r/n=/r/n18/r/n,/r/n得/r/nx/r/n=/r/n9/r/n—/r/n》/r/ny/r/n./r/nT/r/nx/r/n>0/r/n,・・./r/n0v/r/ny/r/nv6/r/n,/r/nS/r/n=/r/nxy/r/n=/r/ny/r/n(/r/n9/r/n—/r/n2/r/ny/r/n)/r/n=/r/n|/r/ny/r/n(6/r/n—/r/ny/r/n)/r/n・/r/n*.*0</r/ny/r/n<6/r/n,・・/r/n6/r/n—/r/ny/r/n>0./r/n

/r/n・・・sw|/r/n(/r/n・・・sw|/r/n(/r/n6/r/n_/r/ny/r/n)/r/n+/r/ny/r/n27/r/nr/r/n当且仅当/r/n6/r/n—/r/ny/r/n=/r/ny/r/n,/r/n即/r/ny/r/n=/r/n3/r/n时/r/n，/r/n等号成立/r/n，/r/n此时/r/nx/r/n=/r/n4.5./r/n故每间虎笼长为/r/n