å©ç¨ææ ¼ææ¥æ°ä¹æ³æ±è¡¨é¢ç§¯ä¸ºa^2çé¿æ¹ä½çæ大ä½ç§¯
设é¿æ¹ä½é¿ä¸ºx,宽为y,é«ä¸ºz
ç®æ å½æ°f(x,y,z)=xyz
éå¶æ¡ä»¶ä¸ºg(x,y,z)=2(xy+yz+xz)=a²
å³Ï(x,y,z)=2(xy+yz+xz)-a²=0
å¼å
¥ææ ¼ææ¥ä¹åλ,æé ææ ¼ææ¥å½æ°L(x,y,z)=f(x,y,z)+λÏ(x,y,z)=xyz+λ[2(xy+yz+xz)-a²]
å
L'x(x,y,z)=yz+2λ(y+z)=0.(1)
L'y(x,y,z)=xz+2λ(x+z)=0.(2)
L'z(x,y,z)=xy+2λ(x+y)=0.(3)
Ï(x,y,z)=2(xy+yz+xz)-a²=0.(4)
ç±(1)(2)(3)å¾
x=y=z=4λ
代å
¥(4)å¾
λ=a/â96=â6a/24
å³é©»ç¹ä¸ºP(x,y,z)=P( â6a/24,â6a/24,â6a/24)
å¯ä¸é©»ç¹,æ
æå¼
æ大ä½ç§¯V=xyz=8λ^3=â6a^3/2304
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