求f(x,y)=xy的驻点,并讨论函数在驻点处是否取得极值
首先对f(x,y)=xy求偏导,最先对x求偏导得到导数y,然后对y求偏导得到导数x令导数y=0；导数x=0
讨论x=0;y=0
对f(x,y)=xy求二阶偏导先对x求二阶偏导得到导数1,然后对y求二阶偏导得到导数1
在根据A的平方-B×C＞0得到f(x,y)=xy在（0,0）点取得极大值.
绛旓細∂z/∂x=y ∂z/∂y=x 鈭达紙0,0锛夋槸椹荤偣 A=∂²z/∂x²=0 B=∂²z/∂x∂y=1 C=∂²z/∂y²=0 P=B²-AC>0 鈭(0,0)涓嶆槸鏋佸肩偣銆
绛旓細瑙ｏ細f'x=y(a-2x-y)=0, 寰梱=0, 鎴杫=a-2x f'y=x(a-x-2y)=0, 寰梮=0, 鎴杧=a-2y 椹荤偣鏈(0, 0), (0, a), (a, 0), (a/3, a/3)A=f"xx=-2y B=f"xy=a-2x-2y C=f"yy=-2x B²-AC=(a-2x-2y)²-4xy 褰搙=0鎴杫=0鏃讹紝鏈塀²-AC>0, ...
绛旓細椹荤偣涓嶆槸鎷愮偣锛岄氬父绉板鏁扮瓑浜0鐨勭偣涓哄嚱鏁鐨勯┗鐐锛堟垨绋冲畾鐐癸紝涓寸晫鐐癸級f(x,y)=xy,鑲畾涓嶆槸鎶涚墿绾 濡傛灉鏄姏鐗╅潰鐨勮瘽 鏂圭▼搴旇鏄痜(x,y)=x^2+y^2
绛旓細寰椹荤偣(3, 2), 锛0,0) , (0, 4), (6, 0), (6, 4)A=f"xx=-2(4y-y²)B=f"xy=(6-2x)(4-2y)=4(3-x)(2-y)C=f"yy=-2(6x-x²)鍦(3,2), A=-8, B=0, C=-18, B²-AC=-144<0, 姝や负鏋佸ぇ鍊肩偣锛屾瀬澶у间负f(3,2)=36;鍦(0,0), A=...
绛旓細寰椹荤偣(3, 2), 锛0,0) , (0, 4), (6, 0), (6, 4)A=f"xx=-2(4y-y²)B=f"xy=(6-2x)(4-2y)=4(3-x)(2-y)C=f"yy=-2(6x-x²)鍦(3,2), A=-8, B=0, C=-18, B²-AC=-144<0, 姝や负鏋佸ぇ鍊肩偣锛屾瀬澶у间负f(3,2)=36;鍦(0,0), A=...
绛旓細绗竴涓閫塂,浠f(x,y)=x^4+y^4-x^2-2xy-y^2 鍒嗗埆姹俧(x,y)瀵箈鐨勫亸瀵兼暟鍜屽y鐨勫亸瀵兼暟銆傝仈绔嬩袱涓亸瀵兼暟寮忓瓙寰楀埌涓変釜椹荤偣(0,0),(1,1),(-1,-1).鍐嶅垎鍒眰A=f(x,y)瀵xx鐨勪簩闃跺亸瀵兼暟锛孊=f(x,y)瀵xy鐨浜岄樁鍋忓鏁帮紝C=f(x,y)瀵yy鐨勪簩闃跺亸瀵兼暟锛岀敤B^2-AC鍒嗗埆甯﹀叆涓変釜...
绛旓細閭ｄ箞鏍规嵁鎷夋牸鏈楁棩涔樻暟娉曪紝鍙煡瑕佹眰z=xy鐨鏈澶у硷紝闇鍏姹侳(x,y)鐨勬瀬鍊肩偣銆傚垎鍒F(x,y)鍑芥暟鐨剎鍜寉姹傚锛屽苟姹傚嚭瀵兼暟涓洪浂鏃剁殑鐐癸紝鍙緱锛屜咶(x,y)/蠁x=y+a=0 蠁F(x,y)/蠁y=x+a=0 鍙坸+y-1=0 閫氳繃鏂圭▼缁勫彲姹傚緱锛宎=-1/2锛寈=1/2锛寉=1/2 閭ｄ箞褰搙=1/2锛寉=1/2鏃讹紝z=xy...
绛旓細瑙ｏ細璁f(x,y)=z=x²锛媦²锛1 鍒欐湁:∂z/∂x=2x=0锛寈=0锛∂z/∂y=2y=0锛寉=0锛岃В寰椾竴涓椹荤偣锛氾紙0锛0锛変护锛欰=∂²z/∂x²=2 B=∂²z/∂x∂y=0 C=∂²z/∂y²=2...
绛旓細涓嶆眰閲囩撼锛岄珮鏁伴棶棰樻垜鍙敤鍒濈瓑鏁板瑙ｅ喅锛屽彧渚涚綉鍙嬩滑娆ｈ祻锛氫緷棰樻剰锛屽彲璁 x锛1+cos胃锛2cos²(胃/2),y锛漵in胃锛2sin(胃/2)cos(胃/2).鈭f(x,y)锛漻y 锛漵in(胃/2)cos³(胃/2)鈭磃²(x,y)锛(16/3)路3sin²(胃/2)路cos²(胃/2)路cos²(胃/2)...
绛旓細f(x,y)=x^3-y^3-3xy 鍒嗗埆瀵箈,y姹鍋忓:fx=3x^2-3y fy=-3y^2-3x.浠x=0,fy=0,鍙緱x=0,y=0,鎴杧=-1,y=1杩2涓椹荤偣.鐒跺悗姹備簩闃跺亸瀵:fxx=6x,fxy=-3,fyy=-6y.x=0,y=0鏃,fxx=0,fxy=-3,fyy=0,(-3)^2-0>0,鎵浠(0,0)涓嶆槸鏋佸肩偣;x=-1,y=1鏃,fxx=-6,...
