基本不等式(高一新教材A版必修第一冊)

8/r/n8/r/n基本不等式/r/n

/r/n第/r/n1/r/n課時(shí)基本不等式/r/n自/r/n主預(yù)習(xí)?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，當(dāng)且僅當(dāng)/r/na/r/n=/r/nb/r/n時(shí)，等號成立./r/n基本不等式/r/n(1)/r/n有關(guān)概念：當(dāng)/r/na/r/n,/r/nb/r/n均為正數(shù)時(shí)，把字叫做正數(shù)/r/na/r/n,/r/nb/r/n的算術(shù)平均數(shù)，把/r/nx/Ob/r/n叫做正數(shù)/r/na/r/n,/r/nb/r/n的幾/r/n何平均數(shù)./r/n(2)/r/n不等式：當(dāng)/r/na/r/n,/r/nb/r/n是任意正實(shí)數(shù)時(shí)，/r/na/r/n,/r/nb/r/n的幾何平均數(shù)不大于它們的算術(shù)平均數(shù)，即/r/n価/r/nW/r/n孚,/r/n當(dāng)且僅當(dāng)/r/na/r/n=/r/nb/r/n時(shí)，等號成立./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中等號成立的條件是/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當(dāng)/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時(shí)/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則/r/na/r/n+/r/nb/r/n的最小值為/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當(dāng)且僅當(dāng)/r/na=b=/r/n1/r/n時(shí)取等號/r/n，/r/n故/r/na/r/n+/r/nb/r/n的最小值為/r/n2.]/r/n

/r/n4/r/n.當(dāng)/r/na/r/n,/r/nb/r/ne/r/nR/r/n時(shí)，下列不等關(guān)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根據(jù)耳藝/r/nA/r/nxy/r/n，/r/n三劇/r/n成立的條件判斷/r/n，/r/n知/r/n①②④/r/n錯(cuò)/r/n，/r/n只有/r/n③/r/n正確/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對基本不等式的理解/r/n例/r/n1/r/n】/r/n對基本不等式的理解/r/n例/r/n1/r/n】/r/n給出下面四個(gè)推導(dǎo)過程：/r/n①/r/n???/r/na/r/n、/r/nb/r/n為正實(shí)數(shù)，豬+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其中正確的推導(dǎo)為/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為正實(shí)數(shù)，?：/r/nQ/r/n萬為正實(shí)數(shù)/r/n，/r/n符合基本不等式的條件/r/n，/r/n故/r/n①/r/n的推導(dǎo)正確./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條件/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是錯(cuò)誤的/r/n./r/na/r/na/r/n由/r/nxy/r/nvo/r/n,/r/n得/r/n2/r/n》均為負(fù)數(shù)/r/n，/r/n但在推導(dǎo)過程中將整體/r/nx/r/n+/r/ny/r/n提出負(fù)號后/r/n，/r/n［/r/n―打、/r/n［/r/n―/r/nX/r/n〕/r/n均變?yōu)檎龜?shù)/r/n，/r/n符/r/nyx/r/nyx/r/nk/r/nyj\x/r/n丿/r/n合均值不等式的條件/r/n，/r/n故/r/n③/r/n正確/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反映了兩個(gè)正數(shù)的和與積之間的關(guān)系/r/n./r/n2/r/n./r/n對基本不等式的準(zhǔn)確掌握要抓住以下兩個(gè)方面：/r/n（/r/n1）/r/n定理成立的條件是/r/na/r/n、/r/nb/r/n都是正數(shù)/r/n./r/n（/r/n2）/r/n“/r/n當(dāng)且/r/n僅當(dāng)/r/n"/r/n的含義：當(dāng)/r/na=b/r/n時(shí)/r/n，/r/n，\/ObW^2^/r/n的等號成立/r/n，/r/n即/r/na=b^^2~/r/n=\//r/n0b/r/n;/r/n僅當(dāng)/r/na=b/r/n時(shí)，空/三、/r/n'Ob/r/n的等號成立/r/n，/r/n即/r/na/r/n的等號成立/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跟蹤訓(xùn)練/r/n1/r/n．下列不等式的推導(dǎo)過程正確的是/r/n①若/r/nx/r/n>/r/nl/r/n,/r/n則/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則/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則/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中忽視了基本不等式等號成立的條件/r/n，/r/n當(dāng)/r/nx/r/n=/r/n丄時(shí)即/r/nx/r/n=/r/n1/r/n時(shí)/r/n，/r/nx/r/n+~/r/n^/r/n2/r/n等號成立/r/n，/r/n因?yàn)?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中忽視了利用基本不等式時(shí)每一項(xiàng)必須為正數(shù)這一條件/r/n./r/n]/r/nx/r/n利用基本不等式比較大小/r/n興空/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則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是兩兩不等的實(shí)數(shù)，則/r/np/r/n=/r/na/r/n2/r/n+/r/nb/r/n2/r/n+/r/nc/r/n2/r/n與/r/nq/r/n=/r/nab/r/n+/r/nbc/r/n+/r/nca/r/n的大小關(guān)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在理解基本不等式時(shí)/r/n，/r/n要從形式到內(nèi)含中理解/r/n，/r/n特別要關(guān)注條件/r/n．/r/n2/r/n./r/n運(yùn)用基本不等式比較大小時(shí)應(yīng)注意成立的條件/r/n，/r/n即/r/na/r/n+/r/nb/r/n±/r/n2\Gb/r/n成立的條件是/r/na/r/n>0/r/n,/r/nb/r/n>0/r/n,/r/n等號成立的條件是/r/na=b；a/r/n2/r/n+b/r/n2/r/n三/r/n2/r/nab/r/n成立的條件是/r/na/r/n,/r/nb/r/n^R,/r/n等號成立的條件是/r/na=b/r/n./r/n跟蹤訓(xùn)練/r/n2/r/n./r/n女口果/r/n0/r/n</r/n跟蹤訓(xùn)練/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的大小順序是/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顯然竺筍/r/n>、莎,/r/n又因?yàn)樘m筍/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利用基本不等式證明不等式/r/n【例/r/n3/r/n】/r/n已知/r/na/r/n,/r/nb/r/n,/r/nc/r/n是互不相等的正數(shù)，且/r/na/r/n+/r/nb/r/n+/r/nc/r/n=/r/n1/r/n,/r/n求證：/r/n1/r/n+/r/n|/r/n+/r/n1/r/n>9./r/nabc/r/n［/r/n思路點(diǎn)撥/r/n］/r/n看到/r/n1/r/n+/r/n1/r/n+/r/n1>9/r/n,/r/n想到將/r/n“/r/n1/r/n”/r/n換成/r/n“/r/na/r/n+/r/nb/r/n+/r/nc/r/n”，/r/n裂項(xiàng)構(gòu)造基本不等式的形式/r/n，/r/n用基本/r/n不等式證明/r/n./r/n［/r/n證明/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當(dāng)且僅當(dāng)/r/na/r/n=/r/nb/r/n=/r/nc/r/n時(shí)取等號/r/n,/r/n?.-/r/n+/r/nb/r/n+/r/n->9./r/nabc/r/n母題探究］/r/n本例條件不變，求證：/r/n［/r/n十/r/n―/r/n1/r/n需—/r/nl/r/nj/r/n>8./r/n［/r/n證明/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當(dāng)且僅當(dāng)/r/na/r/n—/r/nb/r/n—/r/nc/r/n時(shí)取等號,/r/n條件不等式的證明/r/n，/r/n要將待證不等式與已知條件結(jié)合起來考慮/r/n，/r/n比如本題通過/r/n“/r/n1/r/n”/r/n的代換/r/n，/r/n將不等式的左邊化成齊次式/r/n,/r/n一方面為使用基本不等式創(chuàng)造條件/r/n,/r/n另一方面可實(shí)現(xiàn)約分與不等式的右邊建立聯(lián)系/r/n./r/n先局部運(yùn)用基本不等式/r/n,/r/n再利用不等式的性質(zhì)/r/n(/r/n注意限制條件/r/n)/r/n,/r/n通過相加/r/n(/r/n乘/r/n)/r/n合成為待證的不等式既是運(yùn)用基本不等式時(shí)的一種重要技能/r/n,/r/n也是證明不等式時(shí)的一種常用方法/r/n./r/n四/r/n跟蹋訓(xùn)練/r/n已知/r/na/r/n,/r/nb/r/n,/r/nc/r/n^/r/nR/r/n，求證：/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證明/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從而/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萬/r/n—/r/n1/r/n，/r/n求證：/r/na/r/n+/r/n2/r/nb/r/n±/r/n2/r/n、/r/nj6/r/n+/r/n7./r/n［/r/n證明/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則卄/r/n2/r/n方=卄產(chǎn)=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當(dāng)/r/na/r/n—/r/n1/r/n=/r/n—/r/n1/r/n時(shí)/r/n,/r/n即/r/na/r/n=/r/n1/r/n+/r/n\/r/n；/r/n6/r/n時(shí)/r/na/r/n—/r/n1/r/na/r/n＋/r/nb/r/n1/r/n.應(yīng)用基本不等式時(shí)要時(shí)刻注意其成立的條件，只有當(dāng)/r/na/r/n>0/r/n,/r/nb/r/n>0/r/n時(shí)，才會/r/n有価/r/nWp/r/n対于“當(dāng)且僅當(dāng)/r/n時(shí)，'二'成立…”這句話要從兩個(gè)方面理解：一方面，當(dāng)/r/na/r/n=/r/nb/r/n時(shí)，/r/n=莎/r/n;另一方面：當(dāng)仝孑=\厲/r/nb/r/n時(shí)，也有/r/na/r/n=/r/na/r/n＋/r/nb/r/n2/r/n．應(yīng)用基本不等式證明不等式的關(guān)鍵在于進(jìn)行“拼”、“湊”、“拆”、“合”、“放縮”等變形構(gòu)造出符合基本不等式的條件結(jié)構(gòu)/r/n../r/n當(dāng)堂達(dá)標(biāo)。/r/n1/r/n．/r/n思考辨析/r/nTOC\o"1-5"\h\z/r/n(1)/r/n對任意/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，貝/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貝/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當(dāng)/r/na/r/n，/r/nb/r/n都為正數(shù)時(shí)/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只有當(dāng)/r/na>0/r/n時(shí)/r/n，/r/n根據(jù)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?yàn)?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.設(shè)/r/na/r/n>/r/nb/r/n>0/r/n，/r/n則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中等號成立的條件是/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由基本不等式知等號成立的條件為/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設(shè)/r/na/r/n>0/r/n,/r/nb/r/n>0/r/n,/r/n證明：萬+萬/r/n2/r/na/r/n+/r/nb/r/n-/r/n[/r/n證明/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分層作業(yè)基本不等式/r/n（/r/n建議用時(shí)：/r/n60/r/n分鐘/r/n）/r/n［/r/n合格基礎(chǔ)練/r/n］/r/n一/r/n、/r/n選擇題/r/n1/r/n.設(shè)/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則/r/nt/r/n與/r/ns/r/n的大小關(guān)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.下列不等式中正確的是/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則/r/na/r/n+-/r/n2/r/n4/r/n不成立/r/n，/r/n故/r/nA/r/n錯(cuò);/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錯(cuò);/r/na=/r/n4/r/n，/r/nb=/r/n16/r/n，/r/n則/r/n\/r/n；/r/nGbV/r/n竺$，故/r/nC/r/n錯(cuò);/r/n由基本不等式可知/r/nD/r/n項(xiàng)正確/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則下列不等式中錯(cuò)誤的是/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正確，由重要不等式知/r/nB/r/n正確，由/r/n三/r/nab/r/n，??/r/nab=(a/r/n+/r/nb/r/n丿/r/n故選/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則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項(xiàng)正確/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，則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錯(cuò);/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錯(cuò)；/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錯(cuò);/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正確/r/n?/r/n]/r/n二/r/n、/r/n填空題/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貝燈/r/n(/r/na/r/n—/r/nb/r/n)(/r/nb/r/n—/r/nc/r/n丿/r/n與/r/n一/r/n^/r/n-/r/n的大小關(guān)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.某工廠第一年的產(chǎn)量為/r/nA/r/n,/r/n第二年的增長率為/r/na/r/n,/r/n第三年的增長率為/r/nb/r/n則這兩年的平均增長率/r/nx/r/n與增長率的平均值孚的大小關(guān)系為./r/nxW/r/n—/r/n2—/r/n[/r/n用兩種方法求出第三年的產(chǎn)量分別為/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則有/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.當(dāng)且僅當(dāng)/r/na=b/r/n時(shí)等號成立/r/n./r/n]/r/n8/r/n.已知函數(shù)/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時(shí)取得最小值，則/r/na/r/n=/r/n當(dāng)且僅當(dāng)/r/n4/r/n當(dāng)且僅當(dāng)/r/n4/r/nx/r/n=/r/na/r/n,/r/n即/r/nx/r/n=/r/n2/r/na/r/n時(shí)等號成立/r/n，/r/n此時(shí)/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時(shí)/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解答題/r/n9/r/n.已知/r/n9/r/n.已知/r/na/r/n，/r/nb/r/n，/r/nc/r/n為正實(shí)數(shù)，且/r/na/r/n+/r/nb/r/n=/r/nl./r/n求證:/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證明/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當(dāng)且僅當(dāng)/r/na/r/n=/r/nb/r/n時(shí)/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為正數(shù)，求證：/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證明/r/n］/r/n左邊/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為正數(shù)/r/n,/r/nba/r/n???/r/na/r/n+/r/nb/r/n±/r/n2/r/n(/r/n當(dāng)且僅當(dāng)/r/na/r/n=/r/nb/r/n時(shí)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當(dāng)且僅當(dāng)/r/na/r/n=/r/nc/r/n時(shí)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當(dāng)且僅當(dāng)/r/nb=c/r/n時(shí)取/r/n“/r/n=/r/n”/r/n）/r/n./r/n從而/r/n*6（/r/n當(dāng)且僅當(dāng)/r/na/r/n=/r/nb/r/n=/r/nc/r/n時(shí)取等號）./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淪/r/n[/r/n等級過關(guān)練/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項(xiàng)中當(dāng)/r/nx/r/n<0/r/n時(shí)/r/n，/r/nx/r/n+丄/r/n<0v2,/r/nx/r/n?/r/nA/r/n錯(cuò)誤/r/n．/r/nx/r/n2/r/n+/r/n2/r/n?/r/nB/r/n正確/r/n．/r/nB/r/n項(xiàng)中，袞/r/n+r/r/n、/r/nx/r/n2/r/n+/r/n2/r/n?/r/nB/r/n正確/r/n．/r/nx/r/n2/r/n+/r/n3/r/n3/r/n當(dāng)/r/nx/r/n=/r/n0/r/n時(shí)/r/n，/r/n卞百=/r/n2<2/r/n,/r/n顯然選項(xiàng)/r/nC/r/n不正確./r/n4/r/nD/r/n項(xiàng)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錯(cuò)誤/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貝肚/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正確；對于/r/n③，/r/n當(dāng)/r/na/r/n=/r/nb/r/n=T/r/n時(shí)/r/n，/r/n不等式的左邊為丁=-/r/n1/r/n,/r/n右邊為均=/r/n可知/r/n③/r/n不正確；令/r/na/r/n=/r/n1/r/n,/r/nb/r/n=-/r/n1/r/n可知/r/n④/r/n不正確/r/n./r/n]/r/n5/r/n.已知/r/na/r/n、/r/nb/r/n、/r/nc/r/n為不全相等的正實(shí)數(shù)，求證：/r/na/r/n+/r/nb/r/n+/r/nc>"Jab+“Jbc+“Jca/r/n./r/n[/r/n證明/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等號不成立/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，則/r/nxy/r/n的最大值為/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當(dāng)且僅當(dāng)/r/nx/r/n=/r/ny/r/n時(shí)取/r/n“/r/n=/r/n"./r/n]/r/n4/r/n.設(shè)/r/na/r/n,/r/nb/r/n為非零實(shí)數(shù)，給出不等式：/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滬/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正確;/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課時(shí)^基本不等式的應(yīng)用/r/n自主預(yù)習(xí)。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都是正數(shù)，/r/nS/r/n2/r/n⑴若/r/nx/r/n+/r/ny/r/n=/r/nS/r/n（/r/n和為定值/r/n）/r/n，則當(dāng)/r/nx/r/n=/r/ny/r/n時(shí)，積/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積為定值/r/n）/r/n，則當(dāng)/r/nx/r/n=/r/ny/r/n時(shí)，和/r/nx/r/n+/r/ny/r/n取得最小值/r/n2\?./r/n上述命題可歸納為口訣：積定和最小，和定積最大./r/n」初試身手/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則/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豐/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萬無=/r/n2/r/n[/r/n當(dāng)且僅當(dāng)/r/n￥=尋，/r/n即/r/nb=/r/n2/r/na/r/n時(shí)/r/n，/r/n等號成立）/r/n1/r/n4/r/n9/r/n故/r/ny/r/n=/r/na/r/n+/r/nb/r/n的最小值為/r/n2/r/n」/r/n2/r/n2/r/n.若/r/nx/r/n>0/r/n，/r/n則/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飛;/r/n'x/r/n?-/r/n=/r/n2\/2/r/n，/r/n當(dāng)且僅當(dāng)/r/nx/r/n=j/r/n2/r/n時(shí)/r/n，/r/n等號成立/r/n./r/n]/r/nx/r/n\t/r/nx/r/n設(shè)/r/nx/r/n，/r/ny/r/n^/r/nN/r/n*/r/n滿足/r/nx/r/n+/r/ny/r/n=/r/n20/r/n，則/r/nxy/r/n的最大值為/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合作探究毎素養(yǎng)/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興空/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思路點(diǎn)撥/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想到如何才能出現(xiàn)乘積定值；/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需要出現(xiàn)和為定值/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當(dāng)且僅當(dāng)/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時(shí)/r/n，/r/n上式等號成立/r/n，/r/n故當(dāng)/r/nx/r/n—/r/n1/r/n時(shí)/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??當(dāng)且僅當(dāng)/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時(shí)/r/n，/r/ny/r/nmax/r/n=/r/n16/r/n-/r/n觀律方法/r/n利用基本不等式求最值的關(guān)鍵是獲得滿足基本不等式成立條件/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解題時(shí)應(yīng)對照已知和欲求的式子運(yùn)用適當(dāng)?shù)?r/n“/r/n拆項(xiàng)/r/n、/r/n添項(xiàng)/r/n、/r/n配湊/r/n、/r/n變形/r/n”/r/n等方法創(chuàng)設(shè)應(yīng)用基本不等式的條件/r/n./r/n具體可歸納為三句話：若不正/r/n，/r/n用其相反數(shù)/r/n，/r/n改變不等號方向；若不定應(yīng)湊出定和或定積；若不等/r/n，/r/n一般用后面第三章/r/n§3.2/r/n函數(shù)的基本性質(zhì)中學(xué)習(xí)/r/n./r/n跟蹤訓(xùn)練/r/n1/r/n./r/n(1)/r/n已知/r/n跟蹤訓(xùn)練/r/n1/r/n./r/n(1)/r/n已知/r/nx/r/n>0/r/n，/r/n求函數(shù)/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求函數(shù)/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當(dāng)且僅當(dāng)/r/nx/r/n=/r/nx/r/n即/r/nx/r/n=/r/n2/r/n時(shí)等號成立./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的最小值為/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當(dāng)且僅當(dāng)/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時(shí)/r/n，/r/n等號成立./r/n當(dāng)/r/nx/r/n=/r/n6/r/n時(shí)/r/n，/r/n函數(shù)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當(dāng)且僅當(dāng)/r/nx/r/n=/r/n當(dāng)且僅當(dāng)/r/nx/r/n=/r/n3/r/n—/r/nx/r/n,/r/n即/r/nx/r/n=/r/n6/r/n時(shí)/r/n，/r/n等號成立.當(dāng)/r/nx/r/n=/r/n6/r/n時(shí)/r/n，/r/n函數(shù)取得最大值吉./r/n利用基本不等式求條件最值/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且滿足/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當(dāng)且僅當(dāng)/r/n'/r/n當(dāng)且僅當(dāng)/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時(shí)/r/n，/r/n等號成立,/r/nl/r/ny/r/n=/r/n3/r/n故當(dāng)/r/ni/r/n=/r/n12/r/n,/r/ny/r/n=/r/n3/r/n時(shí)/r/n，/r/n（/r/ni/r/n+/r/n2/r/ny/r/n）/r/nmin/r/n=/r/n18./r/n［母題探究］/r/n81/r/n81/r/n若把/r/n“8/r/n+/r/ny/r/n=/r/n1/r/n”/r/n改為/r/n“/r/ni/r/n+/r/n2/r/ny/r/n=/r/n1/r/n”/r/n，其他條件不變，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豐/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當(dāng)且僅當(dāng)竿=;時(shí)取等號,/r/n21/r/n結(jié)/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??當(dāng)/r/ni/r/n=/r/n3/r/n，/r/ny/r/n=/r/n6/r/n時(shí)/r/n，/r/ni/r/n+/r/ny/r/n取到最小值/r/n18./r/n1/r/n．/r/n本題給出的方法/r/n，/r/n用到了基本不等式/r/n，/r/n并且對式子進(jìn)行了變形/r/n，/r/n配湊出滿足基本不等式的條件/r/n，/r/n這是經(jīng)常使用的方法/r/n，/r/n要學(xué)會觀察/r/n、/r/n學(xué)會變形/r/n．/r/nb/r/n常見的變形技巧有：/r/n（/r/n1）/r/n配湊系數(shù)；/r/n（/r/n2）/r/n變符號；/r/n（/r/n3）/r/n拆補(bǔ)項(xiàng)/r/n./r/n常見形式有/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/n四/r/n跟蹤觀練/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時(shí)等號成立/r/n./r/n當(dāng)且僅當(dāng)/r/n＜/r/na/r/nb/r/n，/r/n即/r/n＜/r/n寸/r/n時(shí)等號成立/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的最小值為/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當(dāng)且僅當(dāng)/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的最小值為/r/n3/r/n+/r/n2/r/n叮/r/n2./r/n利用基本不等式解決實(shí)際問題/r/n時(shí)/r/n，/r/n等號成立/r/n，/r/n空型/r/n3//r/n【例/r/n3/r/n】如圖，動(dòng)物園要圍成相同面積的長方形虎籠四間，一面可利用原有的墻，其他各面用鋼筋/r/n【例/r/n3/r/n】如圖，動(dòng)物園要圍成相同面積的長方形虎籠四間，一面可利用原有的墻，其他各面用鋼筋/r/n網(wǎng)圍成．現(xiàn)有/r/n36/r/nm/r/n長的鋼筋網(wǎng)材料，每間虎籠的長、寬分別設(shè)計(jì)為多少時(shí)，可使每間虎籠面積最大？/r/n［/r/n解/r/n］/r/n設(shè)每間虎籠長/r/nx/r/nm/r/n,/r/n寬/r/ny/r/nm/r/n,/r/n則由條件知/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設(shè)每間虎籠面積為/r/nS/r/n,/r/n則/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當(dāng)且僅當(dāng)/r/n2/r/nx/r/n=/r/n3/r/ny/r/n時(shí)/r/n，/r/n等號成立/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故每間虎籠長為/r/n4.5m/r/n,/r/n寬為/r/n3/r/nm/r/n時(shí)/r/n，/r/n可使每間虎籠面積最大./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當(dāng)且僅當(dāng)/r/n6/r/n—/r/ny/r/n=/r/ny/r/n,/r/n即/r/ny/r/n=/r/n3/r/n時(shí)/r/n，/r/n等號成立/r/n，/r/n此時(shí)/r/nx/r/n=/r/n4.5./r/n故每間虎籠長為/r/n