2024·全国·模拟预测
解题方法
1 . 在解决问题“已知正实数
满足
,求
的取值范围”时,可通过重新组合,利用基本不等式构造关于
的不等式,通过解不等式求范围.具体解答如下:
由
,得
,即
,解得
的取值范围是
.
请参考上述方法,求解以下问题:
已知正实数
满足
,则
的取值范围是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb975603433961a27ff01c734d39575f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f29d5f376c75c41ae6af0c8a8565449.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f29d5f376c75c41ae6af0c8a8565449.png)
由
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d14a76fbd7733394b3a7a8c7508ae8f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09a2ebb75f6dc5ba596a98ccbc2bb9be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef066cf9a851361e923ed40c97b842.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f29d5f376c75c41ae6af0c8a8565449.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eed05aa46ec16ee8f98272565d2a2ed9.png)
请参考上述方法,求解以下问题:
已知正实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb975603433961a27ff01c734d39575f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4e7bf9200b351a259ddfc6c0266129d.png)
您最近一年使用:0次
名校
2 . 根据多元微分求条件极值理论,要求二元函数
在约束条件
的可能极值点,首先构造出一个拉格朗日辅助函数
,其中
为拉格朗日系数.分别对
中的
部分求导,并使之为0,得到三个方程组,如下:
,解此方程组,得出解
,就是二元函数
在约束条件
的可能极值点.
的值代入到
中即为极值.
补充说明:【例】求函数
关于变量
的导数.即:将变量
当做常数,即:
,下标加上
,代表对自变量x进行求导.即拉格朗日乘数法方程组之中的
表示分别对
进行求导.
(1)求函数
关于变量
的导数并求当
处的导数值.
(2)利用拉格朗日乘数法求:设实数
满足
,求
的最大值.
(3)①若
为实数,且
,证明:
.
②设
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4a1d0dba29a77dd111efcde543d6c1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc4c14935585e8fa61d032730867d771.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67b6f154c6b2de5695eb1807b98c2c63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/809615d1f91508e2c6c0cda7e592c479.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/244021f826099b18e31af1143597bba2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb5be11a5e6aaf00b2833930b198b4cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0203b006524305c3d8ee0b6c34cd872b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4a1d0dba29a77dd111efcde543d6c1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc4c14935585e8fa61d032730867d771.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1c3c1ed4fb65ab9505ad8078d8d0fb5.png)
补充说明:【例】求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d7ca0caa9933b7afd4bed2683140a07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aebdee8d81b048b5aa520f7e8ba56ff2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a1e15a54c6122c695239107dd0901bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/244021f826099b18e31af1143597bba2.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b3d9ab2fcf15b94f33cb64f84ed906c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
(2)利用拉格朗日乘数法求:设实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0fffbec1fe851795dfdd448bf0d165.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c45d8122b61de13875003d00c002c5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de725a9fc66f67abbe0015131846a648.png)
(3)①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a14c388e1e2e5a2ff1ccf6caffbee0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd24c686fbaaa68705d654b880481ffe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e778f95c72fec00bfbbc63e6dfd0c460.png)
②设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/497d269c30eec393e3f0e877ddbe2983.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ade042c085bbad8aeaf111b9f4c33408.png)
您最近一年使用:0次
名校
3 . 在面积为
的
中,内角
所对的边分别为
,且
.
(1)若
为锐角三角形,
是关于
的方程
的解,求
的取值范围;
(2)若
且
的外接圆的直径为8,
分别在线段
上运动(包括端点),
为边
的中点,且
,
的面积为
.令
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94a7730bbd0a40b11468fefd56737e05.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bea94a11475793bd320592d10f0bdd7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2681f0d1732bcaa8d6362fbd1fa6caa2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad056c25c0fdcbcc765eb5cbc6093f2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1f24c60248a8f0ae275bc69025c0f5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3efffb3e6a571832b723b3c5795b8e8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72cb97395ebc5ee1b212afb7a97b985c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b5a5d0a6eada3f399b4453777430b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
您最近一年使用:0次