微积分全英微积分全英 (49).pdf

《微积分全英微积分全英 (49).pdf》由会员分享，可在线阅读，更多相关《微积分全英微积分全英 (49).pdf（4页珍藏版）》请在淘文阁 - 分享文档赚钱的网站上搜索。

1、 1 Answer to section 12.9()222.Minimize the square of the distance to the plane,subject to 3240.2,2,2 1,3,2,2,23,22,32402 64 Critical point is,.The na7 77f x y zxyzxyzxyzxyzxyz=+=+=81122ture of the problem indicates that this will give a minimum rather than a maximum.2 648 The least distance to the,

2、7 777f=()()()222222222.Minimize the square of the distance,subject to ,90.2,2,2 2,2,22,2,22,90 Critical point are 0,0,3 if 0,f x y zxyzg x y zx yzxyzxy xzxxyyxzzx yzx=+=+=+=10()()()3633662,1,7 if 0,1.29 3,0 if 0,1.92299 0,0,39,2,1,710,3,069223 so the least distance between the origin and the surface

3、 is 3.4xxfff=2().Finding critical points on the interior,2+30;20 the only critical point on the interior:2,1 on the boundary,2+32,22,ffxyyxxyxyxyxy=23()()()22 93 333 33 Critical points are 0,3 ,22223 3327 2,13,0,39,93,2243 33273 3327,93;so the max value is ,93;224224 the min xyfffff+=+=+()value is 2

4、,13.f=()().2 101 There is no minimum or maximum value on the interior since there are an infinite number of critical points.On the boundary,2 1ffxyxyxyxy=+=+=+=25()2211,2 128 Solving these two equations for leads to 1 or 4,together with the constraint 1416 xxyyxyyxyx+=+=3 2812 1942 1912 1942 19Criti

5、cal points:,.55555528 Respectively,the maximum value is ,=21+4 5 and5512 1942 1912 194 the minimum value is,=,555fff +2 19=0.5()()()229.,1,2,2,2,2,0,0,1,2 The Lagrange equations are 12,2=2,2=2,2,21 Critical points are(1,1,0)and(1,1,1);f x y zg x y zxyh x y zxyxyyz=+=+=2 the maximum value (1,1,0)3,th

6、e minimum value (1,1,1)1.ff=4()()()()()()12121212122313121121212130.,subject to g,1 Then ,1,1,1 ,Therefore,nnnnnnnnnnnx xxx xxx xxxxxx xxx xx x xxx xxg x xxx xxg x xxx=+=a2121212121211,then ;11 The maximum value of is when.1 .From partwe have that 1 Therefore,where 1 If nnnnnnnnniiiixxxxxnxxxnnx xxnx xxxxxnaxa=+=b a 121121,then ,so innnniinnaa aaAA AAnaAa aann=