経済数学入門演習問題　WL07, 2021/11/15

I

$(x,y,z)$が \begin{equation*} \left\{ \begin{array}{ccc} x-2y+z+1&=&0\\ x+y-z-2&=&0\\ \end{array} \right. \end{equation*} を動くとき \begin{equation*} f(x,y,z)=x^2+y^2+z^2 \end{equation*} の極値を求めましょう．可能ならば最小値を求めましょう．

II

次の制約条件付の極値問題の停留点を求めましょう．

(1) \begin{equation*} \left\{ \begin{array}{cccc} g_1(x,y,z)&=&y^2+z^2-1&=0\\ g_2(x,y,z)&=&xz-3&=0 \end{array} \right. \end{equation*} の下で \begin{equation*} w=f(x,y,z)=yz+zx \end{equation*}

(2) \begin{equation*} \left\{ \begin{array}{cccc} g_1(x,y,z)&=&3x+y+z-5&=0\\ g_2(x,y,z)&=&x+y+z-1&=0 \end{array} \right. \end{equation*} の下で \begin{equation*} w=f(x,y,z)=x^2+y^2+z^2 \end{equation*}

(3) \begin{equation*} \left\{ \begin{array}{cccc} g_1(x,y,z)&=&x^2+y^2+z^2-1&=0\\ g_2(x,y,z)&=y&=0 \end{array} \right. \end{equation*} の下で \begin{equation*} w=f(x,y,z)=x+y+z^2 \end{equation*}

III

以下の曲線の$\mathrm{P}_0$における接線の方向ベクトルを 求めましょう．

(1) \begin{equation*} \left\{ \begin{array}{ccccc} g_1(x,y,z)&=&x^2+y^2+z^2&=&0\\ g_2(x,y,z)&=&x-y+2z&=&0 \end{array} \right. \end{equation*} at $\mathrm{P}_0(-\frac 1{\sqrt 3},\frac 1{\sqrt 3},\frac 1{\sqrt 3})$.

(2) \begin{equation*} \left\{ \begin{array}{ccccc} g_1(x,y,z)&=&x^2-y^2-z^2-1&=&0\\ g_2(x,y,z)&=&x+y+z&=&0 \end{array} \right. \end{equation*} at $\mathrm{P}_0(\frac 2{\sqrt 2},-\frac 1{\sqrt 2},-\frac 1{\sqrt 2})$.