【微分積分】7-6-3 ヘッセ行列を用いた極大・極小の判定法｜問題集
1.次の関数のヘッセ行列\(H_f(x,y)\)とヘッシアン\(\det H_f(x,y)\)を答えなさい。
(1)\(f(x,y)=2x^2+y^2-xy-7y\)
\(H_f(x,y)=\begin{pmatrix}4 & -1 \\ -1 & 2\end{pmatrix}\)
\(\det H_f(x,y)=7\)
\(\det H_f(x,y)=7\)
(2)\(\displaystyle f(x,y)=\frac{x}{y^2}+xy\)
\(H_f(x,y)=\begin{pmatrix}0 & \displaystyle -\frac{2}{y^3}+1 \\ \displaystyle -\frac{2}{y^3}+1 & \displaystyle \frac{6x}{y^4}\end{pmatrix}\)
\(\displaystyle\det H_f(x,y)=-\left(1-\frac{2}{y^3}\right)^2\)
\(\displaystyle\det H_f(x,y)=-\left(1-\frac{2}{y^3}\right)^2\)
(3)\(f(x,y)=x^3+y^3-3xy\)
\(H_f(x,y)=\begin{pmatrix}6x & -3 \\ -3 & 6y\end{pmatrix}\)
\(\det H_f(x,y)=36xy-9\)
\(\det H_f(x,y)=36xy-9\)
(4)\(f(x,y)=(x^2+y^2)^2-2(x^2-y^2)\)
\(H_f(x,y)=\begin{pmatrix}12x^2+4y^2-4 & 8xy \\ 8xy & 4x^2+12y^2+4\end{pmatrix}\)
\(\det H_f(x,y)=48x^4+96x^2y^2+16y^4+48x^2-16\)
\(\det H_f(x,y)=48x^4+96x^2y^2+16y^4+48x^2-16\)
次の学習に進もう！