2変数関数の極値

■問題

次の関数の極値を求めよ．

${\displaystyle f\left(x,y\right)=x^3+y^3-12x-27y}$

■答

点 ${\displaystyle \left(2,3\right)}$ で極小値 ${\displaystyle -70}$ をとり，点 ${\displaystyle \left(-2,-3\right)}$ で極大値 ${\displaystyle 70}$をとる．

■ヒント

2変数関数の極値の定理1を使用する．

与えられた関数を ${\displaystyle x,y}$ でそれぞれ偏微分し，連立方程式

${\displaystyle \{\begin{array}{l}{\displaystyle \frac{\partial }{\partial x}f\left(x,y\right)=0}\\ {\displaystyle \frac{\partial }{\partial y}f\left(x,y\right)=0}\end{array}}$

とし，その解${\displaystyle \left(x,y\right)=\left(a,b\right)}$を求める．

更に

${\displaystyle \frac{{\partial }^2}{\partial x^2}f\left(x,y\right)=f_{xx}\left(x,y\right)}$ ，${\displaystyle \frac{{\partial }^2}{\partial y^2}f\left(x,y\right)=f_{yy}\left(x,y\right)}$ ，${\displaystyle \frac{{\partial }^2}{\partial y\partial x}f\left(x,y\right)=f_{xy}\left(x,y\right)}$

をそれぞれ求め

${\displaystyle A=f_{xx}\left(a,b\right),}$ ${\displaystyle D={\left\{f_{xy}\left(a,b\right)\right\}}^2-f_{xx}\left(a,b\right)·f_{yy}\left(a,b\right)}$

を計算して極値を判定する．

■解説

与式を ${\displaystyle x}$ で偏微分（偏導関数の定義より， ${\displaystyle y}$ を定数とみなして ${\displaystyle x}$ で微分）すると

${\displaystyle \frac{\partial }{\partial x}f\left(x,y\right)}$ ${\displaystyle =\frac{\partial }{\partial x}\left(x^3+y^3-12x-27y\right)}$ ${\displaystyle =3x^2-12}$

次に ${\displaystyle f\left(x,y\right)}$ を ${\displaystyle y}$ で偏微分（偏導関数の定義より， ${\displaystyle x}$ を定数とみなして ${\displaystyle y}$ で微分）すると

${\displaystyle \frac{\partial }{\partial y}f\left(x,y\right)}$ ${\displaystyle =\frac{\partial }{\partial y}\left(x^3+y^3-12x-27y\right)}$ ${\displaystyle =3y^2-27}$

両者を連立さる．

${\displaystyle \{\begin{array}{l}3x^2-12=0\cdots \cdots \left(1\right)\\ 3y^2-27=0\cdots \cdots \left(2\right)\end{array}}$

(1)から

${\displaystyle 3x^2-12=0}$，${\displaystyle 3x^2=12}$，${\displaystyle x^2=4}$，${\displaystyle x=\pm 2}$

次に(2)から

${\displaystyle 3y^2-27=0}$，${\displaystyle 3y^2=27}$，${\displaystyle y^2=9}$，${\displaystyle y=\pm 3}$

以上から極値をとる候補は ${\displaystyle \left(2,3\right)}$，${\displaystyle \left(2,-3\right)}$，${\displaystyle \left(-2,3\right)}$，${\displaystyle \left(-2,-3\right)}$ の4点となる．

次に

${\displaystyle \frac{{\partial }^2}{\partial x^2}f\left(x,y\right)=f_{xx}\left(x,y\right)}$ ，${\displaystyle \frac{{\partial }^2}{\partial y^2}f\left(x,y\right)=f_{yy}\left(x,y\right)}$ ，${\displaystyle \frac{{\partial }^2}{\partial y\partial x}f\left(x,y\right)=f_{xy}\left(x,y\right)}$

をそれぞれ求める．

${\displaystyle f_{xx}\left(x,y\right)=}$${\displaystyle \frac{{\partial }^2}{\partial x^2}f\left(x,y\right)}$ ${\displaystyle =\frac{\partial }{\partial x}\left(\frac{\partial }{\partial x}f\left(x,y\right)\right)}$ ${\displaystyle =\frac{\partial }{\partial x}\left(3x^2-12\right)}$ ${\displaystyle =6x}$

${\displaystyle f_{yy}\left(x,y\right)=}$${\displaystyle \frac{{\partial }^2}{\partial y^2}f\left(x,y\right)}$ ${\displaystyle =\frac{\partial }{\partial y}\left(\frac{\partial }{\partial y}f\left(x,y\right)\right)}$ ${\displaystyle =\frac{\partial }{\partial y}\left(3y^2-27\right)}$ ${\displaystyle =6y}$

${\displaystyle f_{xy}\left(x,y\right)=}$${\displaystyle \frac{{\partial }^2}{\partial y\partial x}f\left(x,y\right)}$ ${\displaystyle =\frac{\partial }{\partial y}\left(\frac{\partial }{\partial x}f\left(x,y\right)\right)}$ ${\displaystyle =\frac{\partial }{\partial y}\left(3x^2-12\right)}$ ${\displaystyle =0}$

$\left(x,y\right)=\left(a,b\right)$における$A$，$D$の値は

${\displaystyle A}$${\displaystyle =f_{xx}\left(a,b\right)}$${\displaystyle =6a}$

${\displaystyle D}$${\displaystyle ={\left\{f_{xy}\left(a,b\right)\right\}}^2-f_{xx}\left(a,b\right)·f_{yy}\left(a,b\right)}$${\displaystyle =0^2-6a·6b}$${\displaystyle =-36ab}$

これを元に各点における ${\displaystyle A,D}$ を求める．

●点 ${\displaystyle \left(2,3\right)}$ においては

${\displaystyle A}$${\displaystyle =6·2}$${\displaystyle =12}$

${\displaystyle D}$${\displaystyle =-36·2·3}$${\displaystyle =-216}$

となり ${\displaystyle A>0,D<0}$ から点 ${\displaystyle \left(2,3\right)}$ で極小となる．

この点での値は

${\displaystyle f\left(2,3\right)}$${\displaystyle =2^3+3^3-12·2-27·3}$${\displaystyle =8+27-24-81}$${\displaystyle =-70}$

●点 ${\displaystyle \left(2,-3\right)}$ においては

${\displaystyle A}$${\displaystyle =6·2}$${\displaystyle =12}$

${\displaystyle D}$${\displaystyle =-36·2·\left(-3\right)}$${\displaystyle =216}$

${\displaystyle D>0}$ となり ${\displaystyle \left(2,-3\right)}$ は極値ではない．

●点 ${\displaystyle \left(-2,3\right)}$ においては

${\displaystyle A}$${\displaystyle =6·\left(-2\right)}$${\displaystyle =-12}$

${\displaystyle D}$${\displaystyle =-36·\left(-2\right)·3}$${\displaystyle =216}$

${\displaystyle D>0}$ となり ${\displaystyle \left(-2,3\right)}$ は極値ではない．

●点 ${\displaystyle \left(-2,-3\right)}$ においては

${\displaystyle A}$${\displaystyle =6·\left(-2\right)}$${\displaystyle =-12}$

${\displaystyle D}$${\displaystyle =-36·\left(-2\right)·\left(-3\right)}$${\displaystyle =-216}$

となり， ${\displaystyle A<0,D<0}$ から点 ${\displaystyle \left(-2,-3\right)}$ で極大となる．

この点での値は

${\displaystyle f\left(-2,-3\right)}$${\displaystyle ={\left(-2\right)}^3+{\left(-3\right)}^3-12·\left(-2\right)-27·\left(-3\right)}$${\displaystyle =-8-27+24+81}$${\displaystyle =70}$

以上からこの関数は点 ${\displaystyle \left(2,3\right)}$ で極小値 ${\displaystyle -70}$ をとり，点 ${\displaystyle \left(-2,-3\right)}$ で極大値 ${\displaystyle 70}$ をとる．

　

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最終更新日： 2024年5月28日