University of Florida/Egm6321/f09.team1.gzc/Mtg20

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page20-1 $(1)p.19-3Roll{e}^{\prime }sthm\Rightarrow \mathrm{\exists }{\varsigma }_1\in ]0,1[st{G}^{(1)({\varsigma }_1)}\stackrel{(1)}{=}0$ $Now{G}^{(1)}(0)\stackrel{(2)}{=}0why?$ $(1)p.19-1:G^{(1)}(t)\stackrel{(3)}{=}{e}^{(1)}(t)-5t^{4}e(1)$ $(5)p.18-3:e(t)=A(t)-A_2^L(t)(4)$ $e^{(1)}(t)=A^{(1)}-A_2^{L(1)}(t)(5)$ page20-2 $A(t)=\underset{+t}{\overset{-t}{\int }}-=\underset{k}{\overset{-t}{\int }}-+\underset{t}{\overset{k}{\int }}-$ $A^{(1)}\stackrel{(1)}{=}F(-t)+F(t)$ $k\in ]-t,t[(kisconstant)$ $A_2^{L(1)}(t)\stackrel{(2)}{=}\frac{1}{3}[F(-t)+4F(0)+F(t)]+\frac{t}{3}[F^{(1)}(-t)+F^{(1)}(t)$ $e^{(1)}(0)=A^{(1)}(0)-A_2^{L(1)}(0)=2F(0)-[2F(0)+0]=0(3)$ $(3)and(3)p.20-1\Rightarrow (2)p.20-1$ $G^{(1)}({\xi }_1)=0(1)p.20-1$ $G^{(1)}(0)=0(2)p.20-1$

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page20-3

$Roll{e}^{\prime }stheorem\Rightarrow \mathrm{\forall }{\xi }_2\in ]0,{\xi }_1[st{G}^{(2)}({\xi }_2)=0(1)$

$Again,G^{(2)}(0)\stackrel{(2)}{=}0H{W}^{*}4.3$

$(1)and(2)Roll{e}^{\prime }stheorem\Rightarrow \mathrm{\forall }{\xi }_3\in ]0,{\xi }_2[G^{(2)}({\xi }_2)=0st{G}^{(3)}({\xi }_3)\stackrel{(2)}{=}0$

$(1)p.19-1:G^{(3)}(t){\stackrel{(4)}{e}}^{(3)}(t)-\underset{(\xi )(4)(3)}{\underbrace{t^2}}e(1)$

$e^{(3)}(t)\underset{H{W}^{*}4.4}{=}-\frac{t}{3}[F^{(3)}(t)-F^{(3)}(-t)](5)$

$G^{(3)}({\xi }_3)=-\frac{{\xi }_3}{3}\stackrel{ApplyDMVT}{\overbrace{[F^{(3)}({\xi }_3)-F^{(3)}(-{\xi }_3)]}}-60({\xi }_3{)}^{2}e(1)\underset{from(3)}{\stackrel{(6)}{=}}0$

$\underset{(f)}{\stackrel{DMVT}{=}}-\frac{{\xi }_3}{3}[\underset{{\xi }_3-(-{\xi }_3)}{\underbrace{2{\xi }_3}}{F}^{(4)}({\xi }_4)]-60({\xi }_3{)}^{2}e(1){\xi }_4\in ]-{\xi }_3,{\xi }_3[$