経済学で出る数学

ワークブックでじっくり攻める：問0.16解答例

1. 方程式を解くと，$(x,y)=(2,8), (8,2)$．
   $\left\{ \begin{array}{ccc} \displaystyle \frac{\partial f}{\partial x}(x,y)&=&y\\ \displaystyle \frac{\partial f}{\partial y}(x,y)&=&x \end{array} \right.$ なので， $\left\{ \begin{array}{ccc} \displaystyle \frac{\partial f}{\partial x}(2,8)&=&8\\ \displaystyle \frac{\partial f}{\partial y}(2,8)&=&2 \end{array} \right.$ 　 $\left\{ \begin{array}{ccc} \displaystyle \frac{\partial f}{\partial x}(8,2)&=&2\\ \displaystyle \frac{\partial f}{\partial y}(8,2)&=&8. \end{array} \right.$
   $\left\{ \begin{array}{ccc} \displaystyle \frac{\partial g}{\partial x}(x,y)&=&1\\ \displaystyle \frac{\partial g}{\partial y}(x,y)&=&1 \end{array} \right.$


2. 方程式を解くと，$(x,y)=(4,4)$．
   $\left\{ \begin{array}{ccc} \displaystyle \frac{\partial f}{\partial x}(x,y)&=&y\\ \displaystyle \frac{\partial f}{\partial y}(x,y)&=&x \end{array} \right.$ なので， $\left\{ \begin{array}{ccc} \displaystyle \frac{\partial f}{\partial x}(4,4)&=&4\\ \displaystyle \frac{\partial f}{\partial y}(4,4)&=&4 \end{array} \right.$
   $\left\{ \begin{array}{ccc} \displaystyle \frac{\partial g}{\partial x}(x,y)&=&1\\ \displaystyle \frac{\partial g}{\partial y}(x,y)&=&1 \end{array} \right.$





3. 方程式を解くと，$(x,y)=(1,3), (3,1)$．
   $\left\{ \begin{array}{ccc} \displaystyle \frac{\partial f}{\partial x}(x,y)&=&2x\\ \displaystyle \frac{\partial f}{\partial y}(x,y)&=&2y \end{array} \right.$ なので， $\left\{ \begin{array}{ccc} \displaystyle \frac{\partial f}{\partial x}(1,3)&=&2\\ \displaystyle \frac{\partial f}{\partial y}(1,3)&=&6 \end{array} \right.$ 　 $\left\{ \begin{array}{ccc} \displaystyle \frac{\partial f}{\partial x}(3,1)&=&6\\ \displaystyle \frac{\partial f}{\partial y}(3,1)&=&2. \end{array} \right.$
   $\left\{ \begin{array}{ccc} \displaystyle \frac{\partial g}{\partial x}(x,y)&=&1\\ \displaystyle \frac{\partial g}{\partial y}(x,y)&=&1 \end{array} \right.$


4. 方程式を解くと，$(x,y)=(2,2)$．
   $\left\{ \begin{array}{ccc} \displaystyle \frac{\partial f}{\partial x}(x,y)&=&2x\\ \displaystyle \frac{\partial f}{\partial y}(x,y)&=&2y \end{array} \right.$ なので， $\left\{ \begin{array}{ccc} \displaystyle \frac{\partial f}{\partial x}(2,2)&=&4\\ \displaystyle \frac{\partial f}{\partial y}(2,2)&=&4 \end{array} \right.$
   $\left\{ \begin{array}{ccc} \displaystyle \frac{\partial g}{\partial x}(x,y)&=&1\\ \displaystyle \frac{\partial g}{\partial y}(x,y)&=&1 \end{array} \right.$