这个不等式a+b最小值咋求 这个不等式a+b最小值如何求
a+b\u7684\u6700\u5c0f\u503c\u600e\u4e48\u6c42\uff0c\u5747\u503c\u4e0d\u7b49\u5f0fa+b\u22651/5ab,\u56e0\u4e3a\u662f\u5de6\u5f0f\u2265\u53f3\u5f0f\uff0c\u6240\u4ee5\u8981\u53d6\u53f3\u5f0f\u7684\u6700\u5927\u503c\uff0c\u6839\u636e\u5747\u503c\u4e0d\u7b49\u5f0f\uff0c\u53ef\u5f97\u6b64\u5f0f\uff1a1/5ab\u22641/5[(a+b)/2]²\uff0c\u6240\u4ee5a+b\u2265 [(a+b)/2]²,a+b\u5e94\u8be5\u4e0d\u7b49\u4e8e0\u5426\u5219\u8fd9\u9898\u76ee\u51fa\u7684\u5c31\u6ca1\u610f\u4e49\u4e86\u6240\u4ee5\u4e24\u8fb9\u53ef\u7ea6\u6389\u4e00\u4e2aa+b\uff0c\u7136\u540e\u53ef\u4ee5\u89e3\u51faa+b\u226420\u3002
\u624b\u673a\u515a\u6253\u8fd9\u4e9b\u6570\u5b66\u7b26\u53f7\u4e0d\u5bb9\u6613\u7684\u5462\uff5e\u6240\u4ee5\u6ee1\u610f\u8bb0\u5f97\u91c7\u7eb3\u54e6\uff01
20
\u89e3\u6790\uff1a
//\u5047\u5b9aa>0\uff0cb>0
a+b=ab/5\u2264[(a+b)/2]²/5
20(a+b)\u2264(a+b)²
20\u2264a+b
a+b\u226520
绛旓細鏈夊叧鍒╃敤鍩烘湰涓嶇瓑寮忔眰鏈鍊肩殑闂锛屾湁鏃跺繀椤讳娇鐢1鐨勪唬鎹㈡潵瑙ｅ喅銆備緥锛氬凡鐭>0锛宐>0锛2a锛媌=1锛屾眰2/a锛1/b鐨勬渶灏忓銆傘愯В娉曚竴銆戝洜涓篴銆乥閮芥槸姝ｆ暟锛屽垯2a锛媌鈮2鈭(2ab)锛屽洜2a锛媌=1锛屽垯2鈭(2ab)鈮1锛屽緱锛2ab鈮1/4锛1/(ab)鈮8 鍙堬細(2/a)锛(1/b)鈮2鈭歔2/(ab)]锛岃1/(ab...
绛旓細鍩烘湰涓嶇瓑寮忔眰鏈鍊艰繍鐢ㄥ熀鏈笉绛夊紡姹傛渶鍊肩殑涓夊師鍒欌憼a锛宐涓洪潪璐熷疄鏁;鈶″綋鍜宎+b涓哄畾鍊兼椂锛岀Нab鏈夋渶澶у;褰撶Нab涓哄畾鍊兼椂锛屽拰a+b鏈夋渶灏忓;鈶=b鏃讹紝涓嶇瓑寮忎腑鐨勭瓑鍙锋垚绔嬶紝a鈮燽鏃讹紝涓嶇瓑寮忎腑鐨勭瓑鍙蜂笉鎴愮珛(杩欐椂a+b>2ab锛屾剰鍛崇潃a+b鐨勬渶灏忓涓巃b鐨勬渶澶у煎潎涓嶅瓨鍦)銆傚熀鏈笉绛夊紡鐨勫父瑙佸彉褰㈠叕寮 (...
绛旓細灏嗕袱杈瑰钩鏂癸紝鍋囪鐩哥瓑锛岀敤鍩烘湰涓嶇瓑寮锛屾眰ab鐨鏈灏忓锛屽皢ab甯﹀叆鍘熷紡锛宎鍔燽澶т簬绛変簬20
绛旓細涓姝 A銆丅 閮藉繀椤绘槸姝ｆ暟銆備簩瀹 1銆佸湪A+B涓哄畾鍊兼椂锛屼究鍙互鐭ラ亾A路B鐨勬渶澶у硷紱2銆佸湪A路B涓哄畾鍊兼椂锛屼究鍙互鐭ラ亾A+B鐨勬渶灏忓銆備笁鐩哥瓑 褰撲笖浠呭綋A銆丅鐩哥瓑鏃讹紝绛夊紡鎴愮珛锛涘嵆 1銆 A=B ↔ A+B=2鈭AB锛2銆丄鈮燘 ↔ A+B>2鈭欰B銆
绛旓細鈶￠亣鍒/x-a/+/x-b/鈮鍜/x-a/+/x-b/鈮鍨嬶紝鏈変袱绉嶈В娉曪細锛1)闆剁偣鍒嗘娉曪細鍒╃敤闆剁偣鍒嗗尯闂达紝鏋勯犲垎娈靛嚱鏁版潵姹傝В锛屽拰绗憼绫荤殑鍘熺悊涓鏍凤紱(2)鍒╃敤缁濆鍊肩殑鍑犱綍鎰忎箟鏉ョ洿鎺ユ眰锛屽嵆鏄鍏紡鐩存帴浠ｆ暟瀛/a/-/b/鈮/a卤b/鈮/a/+/b/ (3)鏌タ涓嶇瓑寮锛氭病浠涔堝ソ璇寸殑锛岀洿鎺ユ妸鏁板瓧浠ｅ叆鍏紡鍗冲彲锛...
绛旓細濡傛灉a<0鍒欏嚱鏁版湁鏈澶у硷紝褰搙=-(b/2a)鏃讹紝y鍙栨渶澶у硷紝鏈澶у间负y=(4ac-b^2)/4a 瀵逛簬浜屾鍑芥暟y=a(x-h)^2+k(a涓嶇瓑浜0)锛堣繖涓彨鍋氣滈《鐐瑰紡鈥濓級濡傛灉a>0鍒欏嚱鏁版湁鏈灏忓锛屽綋x=h鏃讹紝y鍙栨渶灏忓硷紝鏈灏忓间负y=k 濡傛灉a<0鍒欏嚱鏁版湁鏈澶у硷紝褰搙=h鏃讹紝y鍙栨渶澶у硷紝鏈灏忓间负y=k 浜屾...
绛旓細涓嶇敤 瑙ｆ瀽锛氬潎鍊涓嶇瓑寮锛歛²+b²鈮2ab 鍏舵湰璐ㄦ槸锛歛²+b²-2ab =(a-b)²鈮0(褰撲笖浠呭綋a=b鏃跺彇绛夊彿)
绛旓細ab=a+b+3澶т簬绛変簬2鏍瑰彿ab+3锛屽嵆ab-2鏍瑰彿ab+3澶т簬绛変簬0锛屼护鏍瑰彿ab绛変簬t锛屽垯ab=t^2 鎵浠ュ緱t^2-2t+3澶т簬绛変簬0,瑙ｅ緱t澶т簬绛変簬3,鎴栬卼灏忎簬绛変簬-1(鑸嶅幓)鎵浠ユ牴鍙穉b澶т簬绛変簬3,鎵浠b澶т簬绛変簬9,褰撲笖浠呭綋a=b=3鏃,鍙栧緱鏈灏忓9
绛旓細涓轰簡鑳芥秷鎺夆垰x锛屽彧鑳界敤涓嶇瓑寮鏉ュ仛锛屾牴鎹笉绛夊紡鐨勬ц川a²+b²鈮2ab锛屽綋a=b鏃跺欒兘鍙栧埌绛夊彿锛屼篃灏辨槸鏈灏忓锛屾墍浠ワ紙a+b锛²=a²+b²+2ab鈮4ab锛岃a=鈭歺锛屽彟涓涓涓篵锛屽師寮忊墺4脳鈭歺脳1/锛2鈭歺锛坸+1锛夛冀=2/锛坸+1锛夛紝褰揳=b鏃跺欏彇鍒扮瓑鍙凤紝涔熷氨鏄垰x=...
绛旓細1/a+1/b =(1/a+1/b)[(a+b)/2]=1/2+a/2b+b/2a+1/2 =a/2b+b/2a+1 鈮2鏍瑰彿涓嬶紙a/2b*b/2a锛+1 =2 褰撲笖浠呭綋a/2b=b/2a 鍗砤=b=1 鏃剁瓑鍙锋垚绔 鏈灏忓鏄2
