Re: 請益

[tsungjen (湛紅)]

※ 引述《diedheart (die)》之銘言：
: find extreme value of
: f(x)=e  -xy   (-xy是在次方...即右上角)
: x^2+4y^2<=1
: 謝謝

f(x,y)=e^(-xy)
g(x,y)=x^2+4y^2

fx=-ye^(-xy)
fy=-xe^(-xy)

gx=2x
gy=8y

By Lagrange Multiplier Thm

fx=a gx  =>  -ye^(-xy)=2ax
fy=a gy  =>  -xe^(-xy)=8ay

(1)x^2+4y^2=1
      y                x                               x= 2^(-1/2),-2^(-1/2)
a= - ----e^(-xy) = - ----e^(-xy)   =>  x^2=4y^2   =>   y= 2^(-1/2),-2^(-1/2)
      2x              8y

fmax=e^(1/2)
fmin=e^(-1/2)

(2)x^2+4y^2<1

solve critical points fx=0 and fy=0

fx=-ye^(-xy)=0  =>  y=0   => fxx=y^2e^(-xy)=0   and  fxy=fyx=(xy-1)e^(-xy)=-1
fy=-xe^(-xy)=0  =>  x=0   => fyy=x^2e^(-xy)=0

   / fxx  fxy /
d= /          / = -1 < 0  => saddle point
   / fyx  fyy /

     (行列式)




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