3 $BA02s>R2p$7$?DjM}$N>ZL@(B

Proof. $\Omega$ $B$,3+=89g$G$"$k$+$i!"(B $\exists\eps>0$ s.t. $B(a;\eps)\subset \Omega$. Taylor $B$NDjM}$+$i!"(B

$$(\forall h\in\R^2: \left\Vert h\right\Vert<\eps) (\exists\theta\in(0,1)) \quad f(a+h) =f(a)+f'(a)h+\dfrac{1}{2}\left(H(a+\theta h)h,h\right)$$

$$=f(a)+\dfrac{1}{2}\left(H(a+\theta h)h,h\right).$$

(i)

$H(a)$ $B$,@5CM$J$i$P!"(B $\forall k\in\{1,\cdots,n\}$ $\det H_k(a)>0$. $f$ $B$,(B $C^2$ $B5i$G!"(B $H(x)$ $B$OO"B3$@$+$i!"(B $\exists\eps'\in(0,\eps)$, ( $\forall h\in\R^n$: $\Vert h\Vert<\eps'$)( $\forall\theta\in(0,1)$) $\det H_k(a+\theta h)>0$. $B$3$N$H$-(B ( $\forall h\in\R^n$: $\Vert h\Vert<\eps'$) ( $\forall\theta\in(0,1)$) $H(a+\theta h)$ $B$O@5CM$G$"$k!#(B $B$f$($K(B $h\ne 0$ $B$J$i$P(B $\left(H(a+\theta h)h,h\right)>0$ $B$G$"$k$+$i!"(B $f(a+h)>f(a)$. $B$f$($K(B $f$ $B$O(B $a$ $B$G695A$N6K>.$G$"$k!#(B

(ii)

(i) $B$HF1MM$K>ZL@$G$-$k$N$G>JN,$9$k!#(B

(iii)

$H(a)$ $B$,ITDjId9f$H$9$k!#(B $\exists \lambda,\mu$: $H(a)$ $B$N8GM-CM$G!"(B $\lambda>0$ $B$+$D(B $\mu<0$. $\exists u,v\in\R^n\setminus\{0\}$ s.t. $H(a)u=\lambda u$, $H(a)v=\mu v$. $0<\left\Vert u\right\Vert<\eps$, $0<\Vert v\Vert<\eps$ $B$H$7$FNI$$(B ($B8GM-%Y%/%H%k$O

$$g(t):=f(a+tu), \quad h(t):=f(a+tv)$$

$B$H$*$/$H!"(B

$$g'(t)=f'(a+t u)u,\quad g''(t)=\left(H(a+t u)u,u\right),$$

$$h'(t)=f'(a+t v)v,\quad h''(t)=\left(H(a+t v)v,v\right),$$

$$g'(0)=f'(a)u=0,\quad g''(0)=\left(H(a)u,u\right)=\lambda(u,u)>0,$$

$$h'(0)=f'(a)v=0,\quad h''(0)=\left(H(a)v,v\right)=\mu(v,v)<0$$

$B$G$"$k!#(B$g(t)$ ($\vert t\vert<1$) $B$O(B $t=0$ $B$G695A$N6K>.!"(B $h(t)$ ($\vert t\vert<1$) $B$O(B $t=0$ $B$G695A$N6KBg$G$"$k!#(B $B$f$($K(B $f$ $B$O(B $a$ $B$G6KCM$r.$H$J$C$?$j6KBg$H$J$C$?$j$9$kF=E@(B ($B0HE@(B, saddle point) $B$G$"$k$+$i(B)$B!#(B $\qedsymbol$

ARRAY(0xff088c) $\qedsymbol$