高中數(shù)學(xué)基本不等式的解法十例

1、高中數(shù)學(xué)基本不等式問(wèn)題求解十例一、基本不等式的基礎(chǔ)形式基本不等式的基礎(chǔ)形式1，其中，當(dāng)且僅當(dāng)時(shí)等號(hào)成立。222abab??abR?ab?2，其中，當(dāng)且僅當(dāng)時(shí)等號(hào)成立。2abab????0ab???ab?3?？疾坏仁剑?，其中，當(dāng)且僅當(dāng)時(shí)等號(hào)成22221122abababab??????????????0ab???ab?立。二、常見(jiàn)問(wèn)題及其處理辦法二、常見(jiàn)問(wèn)題及其處理辦法問(wèn)題問(wèn)題1：基本不等式與最值：基本不等式與最值解題思路：（1）積定和最

2、?。喝羰嵌ㄖ?，那么當(dāng)且僅當(dāng)時(shí)，。其中abab???min2abab????0ab???（2）和定積最大：若是定值，那么當(dāng)且僅當(dāng)時(shí)，，其中。ab?ab???2max2abab????????abR?例題例題1：若實(shí)數(shù)滿(mǎn)足，則的最大值是ab221ab??ab?解析：解析：很明顯，和為定，根據(jù)和定積最大法則可得：，當(dāng)且僅當(dāng)時(shí)取等號(hào)。222222212224abababab??????????????????1ab???變式：變式：函數(shù)的圖象恒

3、過(guò)定點(diǎn)A，若點(diǎn)在直線上，則的最大值為1(01)xyaaa????1mxny??mn______。解析：解析：由題意可得函數(shù)圖像恒過(guò)定點(diǎn)，將點(diǎn)代入直線方程中可得，??11A??11A1mxny??1mn??明顯，和為定，根據(jù)和定積最大法則可得：，當(dāng)且僅當(dāng)時(shí)取等號(hào)。2124mnmn?????????12mn??例題例題2：已知函數(shù)，則取最小值時(shí)對(duì)應(yīng)的的值為_(kāi)_________??2122xxfx?????fxx解析：解析：很明顯，積為定，根

4、據(jù)積定和最小法則可得：，當(dāng)且僅當(dāng)2211222122xxxx??????時(shí)取等號(hào)。21212xxx?????解析：解析：由題意可得，左邊乘以可得：，化簡(jiǎn)可得：141xy??141xy??14441yxxyyx????????????????，很明顯中積為定值，根據(jù)積定和最小的法則可得：1441144yyxxxyxy??????????????????44yxxy?，當(dāng)且僅當(dāng)時(shí)取等號(hào)。故而可得442244yxyxxyxy????24184

5、xyxyxy????????。不等式有解，亦即，亦即1444yxxy???????????????234yxmm??＜2min344ymmx??????????，解得或者，故而可得。2340mm???4m?1m??????14m???????變式：變式：若，，且，則的最小值為_(kāi)_________0x?0y?1222xyxy????43xy?解析：解析：由，化簡(jiǎn)題干條件可得乘以所求內(nèi)容可得：????2243xyxyxy?????14222

6、2xyxy????，化簡(jiǎn)后可得：????1414432222222224322xyxyxyxyxyxyxyxy?????????????????????????，很明顯中二者積為定值，根據(jù)積定和最??422241222432xyxyxyxyxy???????????4222222xyxyxyxy?????小法則可得，當(dāng)且僅當(dāng)????4242222224222222xyxyxyxyxyxyxyxy??????????????4222222