题目
题型:不详难度:来源:
已知实数x,y满足
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061317-72637.png)
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061317-23351.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061318-80934.png)
(3)求
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061318-26313.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061318-21051.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061317-23351.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061318-72418.png)
解析
试题分析:∵实数x,y满足
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061317-72637.png)
∴作出可行域如图所示,并求顶点坐标
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061319-84012.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061319-62400.png)
(1)如图,作直线
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061319-55408.png)
由图可知当该直线经过点B时,即
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061319-99275.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061319-39386.png)
当该直线经过点C时,即
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061320-78273.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061320-32083.png)
∴目标函数
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061317-23351.png)
(2)∵
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061320-17964.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061320-57615.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061321-19265.png)
∴由图知
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061321-54498.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061321-82251.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061321-33583.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061318-80934.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061318-72418.png)
(3)∵
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061318-26313.png)
∴从图中易知可行域中的点B到原点的距离最小,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061322-46351.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061322-41953.png)
(4)∵
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061318-21051.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061322-72676.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061322-77354.png)
在图中作出直线
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061322-77354.png)
∴点B到该直线的距离
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061422-96528.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061423-56541.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061423-18740.png)
点评:解决线性规划问题,关键是正确画出可行域,如果目标函数不是线性的,则需要进行适当转化,比如转化为距离、斜率等.
核心考点
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061259-95347.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061259-64042.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061259-17558.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061255-20587.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061255-30820.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061255-97867.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061255-21478.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061251-49614.png)
A.![]() | B.0 | C.12 | D.20 |
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061240-94224.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061240-74859.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061231-90865.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061231-80696.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061231-73812.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191017/20191017061231-90865.png)
A.0,12 | B.12,0 | C.8,4 | D.7,5 |
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