1.3 $B6KCMLdBj$NDjM}(B

Taylor $B$NDjM}$r(B $k=2$ $B$GMQ$$$k!#(B $f$ $B$,(B $C^2$ $B5i$J$i$P!"==J,>.$5$$(B $\forall h\ne0$ $B$KBP$7$F!"(B

$$f(a+h) =f(a)+\left(d^2 f\right)_a(h) +\frac{1}{2!}\left(d^2 f\right)_{a+\theta h}(h)$$

$$=f(a)+f'(a) h +\sum_{i,j=1}^n\frac{\rd^2 f}{\rd x_i\rd x_j}(a+\theta h)h_ih_j.$$

$f'(a)=0$ $B$H$9$k$H!"(B$f'(a)h=0$ $B$J$N$G(B

$$f(a+h)=f(a)+\sum_{i,j=1}^n\frac{\rd^2 f}{\rd x_i\rd x_j}(a+\theta h)h_ih_j.$$

$\left\Vert h\right\Vert$ $B$,>.$5$$$H$-!"(B $B1&JUBh(B2$B9`$O!VBgBN!W(B $h$ $B$N(B$2$$Bl9g$,$"$k!#(B

$$\left(\forall h: 0<\left\Vert h\right\Vert<\eps\right)\quad \sum_{i,j=1}^n\frac{\rd^2 f}{\rd x_i\rd x_j}(a+\theta h)h_ih_j>0 \quad\Then$$   $f$ $B$O(B $a$ $B$G6K>.(B$$,$$

$$\left(\forall h: 0<\left\Vert h\right\Vert<\eps\right)\quad \sum_{i,j=1}^n\frac{\rd^2 f}{\rd x_i\rd x_j}(a+\theta h)h_ih_j<0 \quad\Then$$   $f$ $B$O(B $a$ $B$G6KBg(B$$.$$

Hesse $B9TNs$ON9TNs$G$"$k!#(B $B$3$l$r;H$&$H!">e$N<0$O(B

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

$B$H=q$1$k!#(B

l9g$O!"$b$C$H>\$7$/D4$Y$J$$$HH=Dj$G$-$J$$!#(B \qed \end{jremark}">