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\begin_body

\begin_layout Standard
\begin_inset Formula $f:\Omega\subseteq\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
\end_inset

 es 
\series bold
dos veces diferenciable
\series default
 o 
\series bold
de clase
\series default
 
\begin_inset Formula ${\cal C}^{2}$
\end_inset

 en 
\begin_inset Formula $a\in\Omega$
\end_inset

 si 
\begin_inset Formula $f$
\end_inset

 es diferenciable en 
\begin_inset Formula ${\cal U}\in{\cal E}(a)$
\end_inset

 y 
\begin_inset Formula $df:{\cal U}\rightarrow{\cal L}(\mathbb{R}^{m},\mathbb{R}^{n})\equiv M_{n\times m}(\mathbb{R})\equiv\mathbb{R}^{nm}$
\end_inset

 (la aplicación que a cada elemento de 
\begin_inset Formula ${\cal U}$
\end_inset

 le asigna un vector en 
\begin_inset Formula $\mathbb{R}^{nm}$
\end_inset

 que contiene, en algún orden, los elementos de la matriz asociada a la
 diferencial del elemento) es diferenciable en 
\begin_inset Formula $a$
\end_inset

.
 Por inducción se define el ser 
\begin_inset Formula $n$
\end_inset

 veces diferenciable o de clase 
\begin_inset Formula ${\cal C}^{n}$
\end_inset

, y el ser infinitamente diferenciable o de clase 
\begin_inset Formula ${\cal C}^{\infty}$
\end_inset

.
\end_layout

\begin_layout Standard
Denotamos la derivada parcial 
\begin_inset Formula $k$
\end_inset

-ésima de la derivada parcial 
\begin_inset Formula $j$
\end_inset

-ésima de la 
\begin_inset Formula $i$
\end_inset

-ésima coordenada de 
\begin_inset Formula $f$
\end_inset

, o la 
\begin_inset Formula $i$
\end_inset

-ésima coordenada de la doble derivada parcial respecto a 
\begin_inset Formula $x_{j}$
\end_inset

 y 
\begin_inset Formula $x_{k}$
\end_inset

, como 
\begin_inset Formula $\frac{\partial^{2}f_{i}}{\partial x_{j}\partial x_{k}}$
\end_inset

, y si 
\begin_inset Formula $j=k$
\end_inset

, también escribimos 
\begin_inset Formula $\frac{\partial^{2}f_{i}}{\partial x_{j}^{2}}$
\end_inset

.
 Si 
\begin_inset Formula $f$
\end_inset

 tiene derivadas parciales segundas 
\begin_inset Formula $\frac{\partial f_{i}}{\partial x_{k}\partial x_{j}}$
\end_inset

 todas continuas en 
\begin_inset Formula $a$
\end_inset

 entonces 
\begin_inset Formula $f$
\end_inset

 es dos veces diferenciable en 
\begin_inset Formula $a$
\end_inset

.
\end_layout

\begin_layout Section
Matriz hessiana
\end_layout

\begin_layout Standard
Del mismo modo que podemos pensar en la diferencial de una función diferenciable
 como 
\begin_inset Formula $df(a):\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
\end_inset

 dada por 
\begin_inset Formula $\vec{u}\mapsto d_{\vec{u}}f(a)$
\end_inset

, llamamos 
\series bold
diferencial segunda
\series default
 de 
\begin_inset Formula $f$
\end_inset

 en 
\begin_inset Formula $a$
\end_inset

 a la aplicación 
\begin_inset Formula $d^{2}f(a):\mathbb{R}^{m}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
\end_inset

 con 
\begin_inset Formula $(\vec{u},\vec{v})\mapsto d_{\vec{v}}d_{\vec{u}}f(a)$
\end_inset

, y vemos que esta es una aplicación bilineal.
\end_layout

\begin_layout Standard
La matriz de 
\begin_inset Formula $d^{2}f(a):\mathbb{R}^{m}\times\mathbb{R}^{m}\rightarrow\mathbb{R}$
\end_inset

, dada por
\begin_inset Formula 
\[
\left(\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}\right)_{ij}
\]

\end_inset

 se denomina 
\series bold
matriz hessiana
\series default
.
 Así, si 
\begin_inset Formula $M$
\end_inset

 es la matriz hessiana de 
\begin_inset Formula $f$
\end_inset

, entonces
\begin_inset Formula 
\[
d^{2}f(a)(\vec{u},\vec{v})=\left(\begin{array}{ccc}
- & \vec{u} & -\end{array}\right)M\left(\begin{array}{c}
|\\
\vec{v}\\
|
\end{array}\right)
\]

\end_inset


\end_layout

\begin_layout Standard
Como 
\series bold
teorema
\series default
, sea 
\begin_inset Formula $f:\Omega\subseteq\mathbb{R}^{2}\rightarrow\mathbb{R}$
\end_inset

 y 
\begin_inset Formula $a=(x_{0},y_{0})\in\Omega$
\end_inset

, si 
\begin_inset Formula $\frac{\partial^{2}f}{\partial x\partial y}$
\end_inset

 y 
\begin_inset Formula $\frac{\partial^{2}f}{\partial y\partial x}$
\end_inset

 están definidas en 
\begin_inset Formula $\Omega$
\end_inset

 y son continuas en 
\begin_inset Formula $a$
\end_inset

, entonces su valor en 
\begin_inset Formula $a$
\end_inset

 coincide.
 Esto significa que la matriz Hessiana es simétrica.
 
\series bold
Demostración:
\series default
 Como 
\begin_inset Formula $\Omega$
\end_inset

 es abierto, existe 
\begin_inset Formula $\varepsilon$
\end_inset

 tal que 
\begin_inset Formula $B_{\infty}(a,\varepsilon)=(x_{0}-\varepsilon,x_{0}+\varepsilon)\times(y_{0}-\varepsilon,y_{0}+\varepsilon)\subseteq\Omega$
\end_inset

.
 Fijamos 
\begin_inset Formula $t\in(x_{0}-\varepsilon,y_{0}+\varepsilon)$
\end_inset

 y 
\begin_inset Formula $s\in(y_{0}-\varepsilon,y_{0}+\varepsilon)$
\end_inset

, y consideramos 
\begin_inset Formula $\Delta_{t,s}\coloneqq f(t,s)-f(t,y_{0})-f(x_{0},s)+f(x_{0},y_{0})$
\end_inset

.
 Si ahora llamamos 
\begin_inset Formula $F_{\overline{s}}(\overline{t})\coloneqq f(\overline{t},\overline{s})-f(\overline{t},y_{0})$
\end_inset

, vemos que 
\begin_inset Formula $F_{\overline{s}}(\overline{t})$
\end_inset

 es derivable con 
\begin_inset Formula $F'_{\overline{s}}(\overline{t})=\frac{\partial f}{\partial x}(\overline{t},\overline{s})-\frac{\partial f}{\partial x}(\overline{t},y_{0})$
\end_inset

 y que entonces 
\begin_inset Formula $\Delta_{t,s}=F_{s}(t)-F_{s}(x_{0})=F'_{\overline{s}}(\xi_{t,s})(t-x_{0})=\left(\frac{\partial f}{\partial x}(\xi_{t,s},s)-\frac{\partial f}{\partial x}(\xi_{t,s},y_{0})\right)(t-x_{0})\overset{\Phi(\overline{s})\coloneqq \frac{\partial f}{\partial x}(\xi_{t,s},\overline{s})}{=}(\Phi(s)-\Phi(y_{0}))(t-x_{0})\overset{\Phi\text{ derivable por hipótesis}}{=}\Phi'(\eta_{t,s})(s-y_{0})(t-x_{0})=\frac{\partial^{2}f}{\partial x\partial y}(\xi_{t,s},\eta_{t,s})(s-y_{0})(t-x_{0})$
\end_inset

.
 Permutando los papeles de las dos coordenadas (definiendo 
\begin_inset Formula $\sigma_{\overline{t}}(\overline{s})\coloneqq f(\overline{t},\overline{s})-f(x,\overline{s})$
\end_inset

) obtenemos que 
\begin_inset Formula $\Delta_{t,s}=\sigma_{\overline{t}}(\overline{s})-\sigma_{\overline{t}}(y_{0})=\frac{\partial^{2}f}{\partial y\partial x}(\hat{\xi}_{t,s},\hat{\eta}_{t,s})(s-y_{0})(t-x_{0})$
\end_inset

.
 Haciendo ahora tender 
\begin_inset Formula $(t,s)$
\end_inset

 a 
\begin_inset Formula $(x_{0},y_{0})$
\end_inset

, por la regla del sandwich 
\begin_inset Formula $(\xi_{t,s},\eta_{ts})$
\end_inset

 y 
\begin_inset Formula $(\hat{\xi}_{t,s},\hat{\eta}_{t,s})$
\end_inset

 también tienden a 
\begin_inset Formula $(x_{0},y_{0})$
\end_inset

, y aplicando la continuidad de las derivadas parciales dobles en 
\begin_inset Formula $a$
\end_inset

, nos queda finalmente que 
\begin_inset Formula $\frac{\partial^{2}f}{\partial x\partial y}(x_{0},y_{0})=\frac{\partial^{2}f}{\partial y\partial x}(x_{0},y_{0})$
\end_inset

.
\end_layout

\begin_layout Section
Desarrollos de Taylor
\end_layout

\begin_layout Standard
Despejando de la definición de diferencial, nos queda que 
\begin_inset Formula $f(a+h)=f(a)+df(a)(h)+o(\Vert h\Vert)$
\end_inset

, lo que podemos interpretar como una aproximación de 
\begin_inset Formula $f(x)$
\end_inset

 cerca de 
\begin_inset Formula $a$
\end_inset

 por un polinomio de grado 1.
 Como 
\series bold
teorema
\series default
, si 
\begin_inset Formula $f:\Omega\subseteq\mathbb{R}^{m}\rightarrow\mathbb{R}$
\end_inset

 es dos veces diferenciable en 
\begin_inset Formula $a\in\Omega$
\end_inset

 entonces 
\begin_inset Formula $f(a+h)=f(a)+df(a)(h)+\frac{1}{2}d^{2}f(a)(h,h)+o(\Vert h\Vert^{2})$
\end_inset

.
 
\end_layout

\begin_layout Standard

\series bold
Demostración:
\series default
 Sea 
\begin_inset Formula $R(h)\coloneqq f(a+h)-f(a)-df(a)(h)-\frac{1}{2}d^{2}f(a)(h,h)$
\end_inset

, y hemos de ver que 
\begin_inset Formula $\lim_{h\rightarrow0}\frac{R(h)}{\Vert h\Vert^{2}}=0$
\end_inset

.
 Como todas las normas en 
\begin_inset Formula $\mathbb{R}^{m}$
\end_inset

 son equivalentes, elegimos 
\begin_inset Formula $\Vert\cdot\Vert_{\infty}$
\end_inset

.
 Usamos el teorema del incremento finito, que afirma que si 
\begin_inset Formula $R$
\end_inset

 es diferenciable y 
\begin_inset Formula $\Vert dR(\xi)\Vert\leq M\forall\xi\in[0,h]$
\end_inset

 entonces 
\begin_inset Formula $\Vert R(h)-R(0)\Vert\leq M\cdot\Vert h-0\Vert$
\end_inset

.
 
\begin_inset Formula $R$
\end_inset

 es diferenciable al ser la suma de 
\begin_inset Formula $f(c+h)$
\end_inset

 y un polinomio de grado máximo 2.
 Para estimar 
\begin_inset Formula $\Vert dR\Vert$
\end_inset

 vemos que 
\begin_inset Formula $R(a)=f(a+h)-f(a)-\sum_{i}\frac{\partial f}{\partial x_{i}}(a)h_{i}-\frac{1}{2}\sum_{i,j}\frac{\partial f}{\partial x_{i}\partial x_{j}}(a)h_{i}h_{j}$
\end_inset

, y usando la 
\begin_inset Formula $\delta$
\end_inset

 de Kronecker,
\begin_inset Formula 
\[
\frac{\partial}{\partial x_{k}}\frac{1}{2}\sum_{i,j}\frac{\partial f}{\partial x_{i}\partial x_{j}}(a)h_{i}h_{j}=\frac{1}{2}\sum_{i,j}\left(\frac{\partial f}{\partial x_{i}x_{j}}(a)\delta_{ik}+\frac{\partial f}{\partial x_{i}\partial x_{j}}(a)\delta_{jk}\right)=\frac{1}{2}\cdot2d\left(\frac{\partial f}{\partial x_{k}}\right)(a)(h)
\]

\end_inset

 Por tanto
\begin_inset Formula 
\[
\frac{\partial R}{\partial x_{k}}=\frac{\partial f}{\partial x_{k}}(a+h)-0-\frac{\partial f}{\partial x_{k}}(a)-d\left(\frac{\partial f}{\partial x_{k}}\right)(a)(h)=:\psi_{k}(h)\Vert h\Vert
\]

\end_inset

donde 
\begin_inset Formula $\lim_{h\rightarrow0}\psi(h)=0$
\end_inset

.
 Como 
\begin_inset Formula $\frac{\partial R}{\partial x_{k}}$
\end_inset

 es continua, definiendo el compacto 
\begin_inset Formula $[0,h]$
\end_inset

 como 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\xout off
\uuline off
\uwave off
\noun off
\color none

\begin_inset Formula $\{th\}_{t\in[0,1]}$
\end_inset


\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\xout default
\uuline default
\uwave default
\noun default
\color inherit
 existe un punto 
\begin_inset Formula $t_{k,h}h\in[0,h]$
\end_inset

 tal que 
\begin_inset Formula 
\[
\frac{\partial R}{\partial x_{k}}(t_{k,h}h)=\max\left\{ \frac{\partial R}{\partial x_{k}}(\xi)\right\} _{\xi\in[0,h]}
\]

\end_inset

Por esto, y como 
\begin_inset Formula $dR(\xi)\equiv\left(\frac{\partial R}{\partial x_{1}}(\xi),\dots,\frac{\partial R}{\partial x_{m}}(\xi)\right)$
\end_inset

, existe 
\begin_inset Formula $C$
\end_inset

 tal que 
\begin_inset Formula 
\[
\Vert dR(\xi)\Vert\leq C\cdot\Vert dR(a)\Vert_{\infty}=C\cdot\max\left\{ \left|\frac{\partial R}{\partial x_{k}}(a)\right|\right\} _{k=1}^{n}
\]

\end_inset

para 
\begin_inset Formula $\xi\in[0,h]$
\end_inset

, y por el teorema del incremento finito, si 
\begin_inset Formula $p$
\end_inset

 es tal que 
\begin_inset Formula 
\[
\left|\frac{\partial R}{\partial x_{p}}(a)\right|=\max\left\{ \left|\frac{\partial R}{\partial x_{k}}(a)\right|\right\} _{k=1}^{n}
\]

\end_inset

tenemos 
\begin_inset Formula 
\[
\Vert R(h)\Vert=\Vert R(h)-R(0)\Vert\leq C\left|\frac{\partial R}{\partial x_{p}}(t_{p,h})\right|\Vert h\Vert=C\psi_{p}(t_{p,h}h)\Vert h\Vert\Vert h\Vert=C\psi_{p}(t_{p,h}h)\Vert h\Vert^{2}
\]

\end_inset

 y entonces 
\begin_inset Formula $\frac{|R(h)|}{\Vert h\Vert^{2}}\leq C\psi_{p}(t_{p,h}h)\rightarrow0$
\end_inset

, lo que prueba el teorema.
\end_layout

\begin_layout Section
Extremos relativos
\end_layout

\begin_layout Standard
Si 
\begin_inset Formula $V$
\end_inset

 es un 
\begin_inset Formula $K$
\end_inset

-espacio vectorial con 
\begin_inset Formula $k\coloneqq \dim_{K}(V)<+\infty$
\end_inset

 y 
\begin_inset Formula $\sigma:V\times V\rightarrow\mathbb{R}$
\end_inset

 una aplicación bilineal, existe 
\begin_inset Formula $A=(a_{ij})\in{\cal M}_{k}(K)$
\end_inset

 asociado a 
\begin_inset Formula $\sigma$
\end_inset

 y podemos definir
\begin_inset Formula 
\[
\Delta_{1}=\left|a_{11}\right|,\Delta_{2}=\left|\begin{array}{cc}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{array}\right|,\dots,\Delta_{k}=\left|\begin{array}{ccc}
a_{11} & \cdots & a_{1k}\\
\vdots & \ddots & \vdots\\
a_{k1} & \cdots & a_{kk}
\end{array}\right|
\]

\end_inset

Entonces un 
\series bold
teorema
\series default
 de álgebra nos dice que 
\begin_inset Formula $\sigma$
\end_inset

 es:
\end_layout

\begin_layout Enumerate
Semidefinida positiva si y sólo si 
\begin_inset Formula $\Delta_{i}\geq0\forall i$
\end_inset

.
\end_layout

\begin_layout Enumerate
Semidefinida negativa si y sólo si 
\begin_inset Formula $\Delta_{i}(-1)^{i}\geq0\forall i$
\end_inset

.
\end_layout

\begin_layout Enumerate
Definida positiva si y sólo si 
\begin_inset Formula $\Delta_{i}>0\forall i$
\end_inset

.
\end_layout

\begin_layout Enumerate
Definida negativa si y sólo si 
\begin_inset Formula $\Delta_{i}(-1)^{i}>0\forall i$
\end_inset

.
\end_layout

\begin_layout Standard
Como 
\series bold
teorema
\series default
, sea 
\begin_inset Formula $\Omega\subseteq\mathbb{R}^{m}$
\end_inset

 abierto, 
\begin_inset Formula $f:\Omega\rightarrow\mathbb{R}$
\end_inset

 y 
\begin_inset Formula $a\in\Omega$
\end_inset

,
\end_layout

\begin_layout Enumerate
Si 
\begin_inset Formula $f$
\end_inset

 alcanza en 
\begin_inset Formula $a$
\end_inset

 un extremo relativo entonces 
\begin_inset Formula $df(a)=0$
\end_inset

.
\begin_inset Newline newline
\end_inset

Podemos suponer que alcanza un máximo.
 Entonces 
\begin_inset Formula $\exists{\cal U}\in{\cal E}(a):f(x)\leq f(a)\forall x\in{\cal U}$
\end_inset

, luego si para 
\begin_inset Formula $i\in\{1,\dots,m\}$
\end_inset

 definimos 
\begin_inset Formula $\varphi_{i}(t)\coloneqq f(a_{1},\dots,a_{i-1},t,a_{i+1},\dots,a_{m})$
\end_inset

, fijado 
\begin_inset Formula $i$
\end_inset

, 
\begin_inset Formula $\exists\varepsilon>0:\forall t\in(a-\varepsilon,a+\varepsilon),\varphi_{i}(t)\leq\varphi_{i}(a_{i})$
\end_inset

 y 
\begin_inset Formula $\frac{\partial f}{\partial x_{i}}(a_{1},\dots,a_{i-1},t,a_{i+1},\dots,a_{m})=\varphi'(t)$
\end_inset

, luego 
\begin_inset Formula $\frac{\partial f}{\partial x_{i}}(a)=\varphi'(a_{i})=0$
\end_inset

 y 
\begin_inset Formula 
\[
df(a)=\left(\begin{array}{ccc}
\frac{\partial f}{\partial x_{1}}(a) & \cdots & \frac{\partial f}{\partial x_{m}}(a)\end{array}\right)=0
\]

\end_inset


\end_layout

\begin_layout Enumerate
Si 
\begin_inset Formula $f$
\end_inset

 es de clase 
\begin_inset Formula ${\cal C}^{2}$
\end_inset

 y 
\begin_inset Formula $df(a)=0$
\end_inset

, entonces
\begin_inset Formula 
\begin{eqnarray*}
d^{2}f(a)\text{ definida positiva} & \implies & f\text{ tiene un mínimo estricto en }a\implies\\
\implies f\text{ tiene un mínimo en }a & \implies & d^{2}f(a)\text{ semidefinida positiva}
\end{eqnarray*}

\end_inset


\end_layout

\begin_deeper
\begin_layout Description
\begin_inset Formula $1\implies2]$
\end_inset

 Consideremos el desarrollo de Taylor de 
\begin_inset Formula $f$
\end_inset

 de orden 2 en 
\begin_inset Formula $a$
\end_inset

, que como 
\begin_inset Formula $df(a)=0$
\end_inset

, queda como
\begin_inset Formula 
\begin{eqnarray*}
f(x) & = & f(a)+\frac{1}{2}d^{2}f(a)(x-a,x-a)+o(\Vert x-a\Vert^{2})\\
 & = & f(a)+\frac{1}{2}\Vert x-a\Vert^{2}d^{2}f(a)\left(\frac{x-a}{\Vert x-a\Vert},\frac{x-a}{\Vert x-a\Vert}\right)+o(\Vert x-a\Vert^{2})\\
 & = & f(a)+\frac{1}{2}\Vert x-a\Vert^{2}\left(d^{2}f(a)\left(\frac{x-a}{\Vert x-a\Vert},\frac{x-a}{\Vert x-a\Vert}\right)+\frac{2o(\Vert x-a\Vert^{2})}{\Vert x-a\Vert^{2}}\right)
\end{eqnarray*}

\end_inset

suponiendo 
\begin_inset Formula $x\neq a$
\end_inset

.
 Pero 
\begin_inset Formula $\frac{x-a}{\Vert x-a\Vert}\in\{y\in\mathbb{R}^{m}\mid \Vert y\Vert=1\}=:K$
\end_inset

, que es compacto por ser cerrado y acotado, y 
\begin_inset Formula $\Phi:\mathbb{R}^{m}\rightarrow\mathbb{R}$
\end_inset

 dada por 
\begin_inset Formula $\Phi(u)\coloneqq d^{2}f(a)(u,u)=\sum_{i,j}\frac{\partial f}{\partial x_{i}\partial x_{j}}(a)u_{i}u_{j}$
\end_inset

 es continua, luego 
\begin_inset Formula $\Phi(K)=\left\{ d^{2}f(a)\left(\frac{x-a}{\Vert x-a\Vert},\frac{x-a}{\Vert x-a\Vert}\right)\right\} _{x\in\mathbb{R}^{m}}$
\end_inset

 es compacto y, por ser además 
\begin_inset Formula $d^{2}f(a)$
\end_inset

 definida positiva, existe 
\begin_inset Formula $M>0$
\end_inset

 tal que 
\begin_inset Formula $\Phi(K)\geq M$
\end_inset

.
\begin_inset Newline newline
\end_inset

Como 
\begin_inset Formula $\frac{2o(\Vert x-a\Vert^{2})}{\Vert x-a\Vert^{2}}\rightarrow0$
\end_inset

 cuando 
\begin_inset Formula $x\rightarrow a$
\end_inset

, existe 
\begin_inset Formula ${\cal U}\in{\cal E}(a)$
\end_inset

 tal que 
\begin_inset Formula $\forall x\in{\cal U},\left|\frac{2o(\Vert x-a\Vert)^{2}}{\Vert x-a\Vert^{2}}\right|<M$
\end_inset

, luego si 
\begin_inset Formula $x\in{\cal U}\backslash\{a\},d^{2}f(a)\left(\frac{x-a}{\Vert x-a\Vert},\frac{x-a}{\Vert x-a\Vert}\right)+\frac{2o(\Vert x-a\Vert^{2})}{\Vert x-a\Vert^{2}}>M-M=0$
\end_inset

 y 
\begin_inset Formula $f(x)>f(a)+\frac{1}{2}\Vert x-a\Vert^{2}\cdot0=f(a)$
\end_inset

.
\end_layout

\begin_layout Description
\begin_inset Formula $2\implies3]$
\end_inset

 Obvio.
\end_layout

\begin_layout Description
\begin_inset Formula $3\implies4]$
\end_inset

 Fijamos 
\begin_inset Formula $u\in\mathbb{R}^{m}$
\end_inset

 y definimos 
\begin_inset Formula $\varphi(t)\coloneqq a+tu$
\end_inset

 como la función 
\begin_inset Formula $\varphi:\mathbb{R}\rightarrow\mathbb{R}^{m}$
\end_inset

 que parametriza la recta 
\begin_inset Formula $a+<\vec{u}>$
\end_inset

.
 Sea 
\begin_inset Formula ${\cal U}\in{\cal E}(a)$
\end_inset

 con 
\begin_inset Formula $f(a)\leq f(x)\forall x\in{\cal U}$
\end_inset

, si restringimos 
\begin_inset Formula $\varphi$
\end_inset

 a 
\begin_inset Formula $\varphi^{-1}({\cal U})$
\end_inset

, un entorno de 0 en 
\begin_inset Formula $\mathbb{R}$
\end_inset

, entonces 
\begin_inset Formula $f\circ\varphi$
\end_inset

 alcanza un mínimo en 0, pues 
\begin_inset Formula $(f\circ\varphi)(0)=f(\varphi(0))=f(a)\leq f(\varphi(t))\forall t\in\varphi^{-1}({\cal U})$
\end_inset

, y tenemos que 
\begin_inset Formula $f\circ\varphi$
\end_inset

 es de clase 
\begin_inset Formula ${\cal C}^{2}$
\end_inset

 y semidefinida positiva.
 Por la regla de la cadena, al ser 
\begin_inset Formula $\varphi$
\end_inset

 y 
\begin_inset Formula $f$
\end_inset

 diferenciables,
\begin_inset Formula 
\begin{multline*}
d(f\circ\varphi)(t)=df(\varphi(t))\circ d\varphi(t)\equiv\\
\equiv\left(\begin{array}{ccc}
\frac{\partial f}{\partial x_{1}}(a+tu) & \cdots & \frac{\partial f}{\partial x_{m}}(a+tu)\end{array}\right)\left(\begin{array}{c}
u_{1}\\
\vdots\\
u_{m}
\end{array}\right)=\sum_{i}\frac{\partial f}{\partial x_{i}}(a+tu)u_{i}
\end{multline*}

\end_inset

Entonces 
\begin_inset Formula $d^{2}(f\circ\varphi)(t)\equiv\frac{d}{dt}\sum_{i}\frac{\partial f}{\partial x_{i}}(a+tu)u_{i}=\sum_{i}u_{i}\frac{d}{dt}\left(\frac{\partial f}{\partial x_{i}}(a+tu)\right)$
\end_inset

.
 Como 
\begin_inset Formula $\frac{\partial f}{\partial x_{i}}$
\end_inset

 es diferenciable al ser 
\begin_inset Formula $f$
\end_inset

 de clase 
\begin_inset Formula ${\cal C}^{2}$
\end_inset

, 
\begin_inset Formula 
\begin{multline*}
\frac{d}{dt}\left(\frac{\partial f}{\partial x_{i}}(a+tu)\right)=d\left(\frac{\partial f}{\partial x_{i}}\circ\varphi\right)(t)=d\frac{\partial f}{\partial x_{i}}(\varphi(t))\circ d\varphi(t)\equiv\\
\equiv\left(\begin{array}{ccc}
\frac{\partial^{2}f}{\partial_{i}\partial_{1}}(a+tu) & \cdots & \frac{\partial^{2}f}{\partial_{i}\partial_{m}}(a+tu)\end{array}\right)\left(\begin{array}{c}
u_{1}\\
\vdots\\
u_{m}
\end{array}\right)=\sum_{j}\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}(a+tu)u_{j}
\end{multline*}

\end_inset

Sustituyendo, 
\begin_inset Formula $d^{2}(f\circ\varphi)(t)\equiv\sum_{i,j}\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}(a+tu)u_{i}u_{j}=d^{2}f(a+tu)(u,u)$
\end_inset

.
 Pero al ser 
\begin_inset Formula $f\circ\varphi$
\end_inset

 una función real de una variable real dos veces derivable con su mínimo
 en 0, sustituyendo 
\begin_inset Formula $0\leq(f\circ\varphi)''(0)=d^{2}f(a)(u,u)$
\end_inset

, y como esto se cumple para todo 
\begin_inset Formula $u\in\mathbb{R}^{m}$
\end_inset

, queda probado que 
\begin_inset Formula $d^{2}f(a)$
\end_inset

 es semidefinida positiva.
\end_layout

\end_deeper
\end_body
\end_document