Càlcul de la Hessiana en una funció de 4 variables

Determina la matriu Hessiana en el punt $(2, -1, 1, -1)$ de la següent funció amb 4 variables:

\begin{equation}
f(x, y, z, w) = 2x^3 y^4 z w^2 – 2y^3 w^4 + 3x^2 z^2
\end{equation}

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Per calcular la matriu Hessiana, seguim aquests passos:

1. Calcular les derivades parciales de primer ordre.
2. Calcular les derivades parciales de segon ordre.
3. Construir la matriu Hessiana amb les derivades de segon ordre.
4. Avaluar la matriu Hessiana en el punt $(2, -1, 1, -1)$.

La matriu Hessiana d’una funció de 4 variables és una matriu simètrica $4 \times 4$, que conté totes les derivades parciales de segon ordre.

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Pas 1: Derivades parciales de primer ordre

La funció és:
\begin{equation}
f(x, y, z, w) = 2x^3 y^4 z w^2 – 2y^3 w^4 + 3x^2 z^2.
\end{equation}

Calculem les derivades parciales respecte a $x$, $y$, $z$ i $w$:

• Derivada respecte a $x$:
  \begin{equation}
  \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left( 2x^3 y^4 z w^2 – 2y^3 w^4 + 3x^2 z^2 \right) = 6x^2 y^4 z w^2 + 6x z^2.
  \end{equation}
  (El terme $-2y^3 w^4$ no depèn de $x$, per tant, la seva derivada és $0$.)
• Derivada respecte a $y$:
  \begin{equation}
  \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left( 2x^3 y^4 z w^2 – 2y^3 w^4 + 3x^2 z^2 \right) = 8x^3 y^3 z w^2 – 6y^2 w^4.
  \end{equation}
  (El terme $3x^2 z^2$ no depèn de $y$.)
• Derivada respecte a $z$:
  \begin{equation}
  \frac{\partial f}{\partial z} = \frac{\partial}{\partial z} \left( 2x^3 y^4 z w^2 – 2y^3 w^4 + 3x^2 z^2 \right) = 2x^3 y^4 w^2 + 6x^2 z.
  \end{equation}
  (El terme $-2y^3 w^4$ no depèn de $z$.)
• Derivada respecte a $w$:
  \begin{equation}
  \frac{\partial f}{\partial w} = \frac{\partial}{\partial w} \left( 2x^3 y^4 z w^2 – 2y^3 w^4 + 3x^2 z^2 \right) = 4x^3 y^4 z w – 8y^3 w^3.
  \end{equation}
  (El terme $3x^2 z^2$ no depèn de $w$.)

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Pas 2: Derivades parciales de segon ordre

La matriu Hessiana requereix les derivades de segon ordre: $\frac{\partial^2 f}{\partial x^2}$, $\frac{\partial^2 f}{\partial y^2}$, $\frac{\partial^2 f}{\partial z^2}$, $\frac{\partial^2 f}{\partial w^2}$, i les derivades mixtes $\frac{\partial^2 f}{\partial x \partial y}$, $\frac{\partial^2 f}{\partial x \partial z}$, $\frac{\partial^2 f}{\partial x \partial w}$, $\frac{\partial^2 f}{\partial y \partial z}$, $\frac{\partial^2 f}{\partial y \partial w}$, $\frac{\partial^2 f}{\partial z \partial w}$. Les derivades mixtes són simètriques ($\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}$, etc.).

a) Derivades respecte a una sola variable

• $\frac{\partial^2 f}{\partial x^2}$:
  \begin{equation}
  \frac{\partial f}{\partial x} = 6x^2 y^4 z w^2 + 6x z^2,
  \end{equation}
  \begin{equation}
  \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x} \left( 6x^2 y^4 z w^2 + 6x z^2 \right) = 12x y^4 z w^2 + 6z^2.
  \end{equation}
• $\frac{\partial^2 f}{\partial y^2}$:
  \begin{equation}
  \frac{\partial f}{\partial y} = 8x^3 y^3 z w^2 – 6y^2 w^4,
  \end{equation}
  \begin{equation}
  \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y} \left( 8x^3 y^3 z w^2 – 6y^2 w^4 \right) = 24x^3 y^2 z w^2 – 12y w^4.
  \end{equation}
• $\frac{\partial^2 f}{\partial z^2}$:
  \begin{equation}
  \frac{\partial f}{\partial z} = 2x^3 y^4 w^2 + 6x^2 z,
  \end{equation}
  \begin{equation}
  \frac{\partial^2 f}{\partial z^2} = \frac{\partial}{\partial z} \left( 2x^3 y^4 w^2 + 6x^2 z \right) = 6x^2.
  \end{equation}
  (El terme $2x^3 y^4 w^2$ no depèn de $z$.)
• $\frac{\partial^2 f}{\partial w^2}$:
  \begin{equation}
  \frac{\partial f}{\partial w} = 4x^3 y^4 z w – 8y^3 w^3,
  \end{equation}
  \begin{equation}
  \frac{\partial^2 f}{\partial w^2} = \frac{\partial}{\partial w} \left( 4x^3 y^4 z w – 8y^3 w^3 \right) = 4x^3 y^4 z – 24y^3 w^2.
  \end{equation}

b) Derivades mixtes

• $\frac{\partial^2 f}{\partial x \partial y}$:
  \begin{equation}
  \frac{\partial f}{\partial x} = 6x^2 y^4 z w^2 + 6x z^2,
  \end{equation}
  \begin{equation}
  \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial}{\partial y} \left( 6x^2 y^4 z w^2 + 6x z^2 \right) = 24x^2 y^3 z w^2.
  \end{equation}
  (El terme $6x z^2$ no depèn de $y$.)
• $\frac{\partial^2 f}{\partial x \partial z}$:
  \begin{equation}
  \frac{\partial f}{\partial x} = 6x^2 y^4 z w^2 + 6x z^2,
  \end{equation}
  \begin{equation}
  \frac{\partial^2 f}{\partial x \partial z} = \frac{\partial}{\partial z} \left( 6x^2 y^4 z w^2 + 6x z^2 \right) = 6x^2 y^4 w^2 + 12x z.
  \end{equation}
• $\frac{\partial^2 f}{\partial x \partial w}$:
  \begin{equation}
  \frac{\partial f}{\partial x} = 6x^2 y^4 z w^2 + 6x z^2,
  \end{equation}
  \begin{equation}
  \frac{\partial^2 f}{\partial x \partial w} = \frac{\partial}{\partial w} \left( 6x^2 y^4 z w^2 + 6x z^2 \right) = 12x^2 y^4 z w.
  \end{equation}
  (El terme $6x z^2$ no depèn de $w$.)
• (\frac{\partial^2 f}{\partial y \partial z}):
  \begin{equation}
  \frac{\partial f}{\partial y} = 8x^3 y^3 z w^2 – 6y^2 w^4,
  \end{equation}
  \begin{equation}
  \frac{\partial^2 f}{\partial y \partial z} = \frac{\partial}{\partial z} \left( 8x^3 y^3 z w^2 – 6y^2 w^4 \right) = 8x^3 y^3 w^2.
  \end{equation}
  (El terme $-6y^2 w^4$ no depèn de $z$.)
• $\frac{\partial^2 f}{\partial y \partial w}$:
  \begin{equation}
  \frac{\partial f}{\partial y} = 8x^3 y^3 z w^2 – 6y^2 w^4,
  \end{equation}
  \begin{equation}
  \frac{\partial^2 f}{\partial y \partial w} = \frac{\partial}{\partial w} \left( 8x^3 y^3 z w^2 – 6y^2 w^4 \right) = 16x^3 y^3 z w – 24y^2 w^3.
  \end{equation}
• $\frac{\partial^2 f}{\partial z \partial w}$:
  \begin{equation}
  \frac{\partial f}{\partial z} = 2x^3 y^4 w^2 + 6x^2 z,
  \end{equation}
  \begin{equation}
  \frac{\partial^2 f}{\partial z \partial w} = \frac{\partial}{\partial w} \left( 2x^3 y^4 w^2 + 6x^2 z \right) = 4x^3 y^4 w.
  \end{equation}
  (El terme $6x^2 z$ no depèn de $w$.)
• Derivades mixtes simètriques:
  \begin{equation}
  \frac{\partial^2 f}{\partial y \partial x} = \frac{\partial^2 f}{\partial x \partial y} = 24x^2 y^3 z w^2,
  \end{equation}
  \begin{equation}
  \frac{\partial^2 f}{\partial z \partial x} = \frac{\partial^2 f}{\partial x \partial z} = 6x^2 y^4 w^2 + 12x z,
  \end{equation}
  \begin{equation}
  \frac{\partial^2 f}{\partial w \partial x} = \frac{\partial^2 f}{\partial x \partial w} = 12x^2 y^4 z w,
  \end{equation}
  \begin{equation}
  \frac{\partial^2 f}{\partial z \partial y} = \frac{\partial^2 f}{\partial y \partial z} = 8x^3 y^3 w^2,
  \end{equation}
  \begin{equation}
  \frac{\partial^2 f}{\partial w \partial y} = \frac{\partial^2 f}{\partial y \partial w} = 16x^3 y^3 z w – 24y^2 w^3,
  \end{equation}
  \begin{equation}
  \frac{\partial^2 f}{\partial w \partial z} = \frac{\partial^2 f}{\partial z \partial w} = 4x^3 y^4 w.
  \end{equation}

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Pas 3: Construcció de la matriu Hessiana

La matriu Hessiana és:
\begin{equation}
H = \begin{bmatrix}
\frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x \partial y} & \frac{\partial^2 f}{\partial x \partial z} & \frac{\partial^2 f}{\partial x \partial w} \\
\frac{\partial^2 f}{\partial y \partial x} & \frac{\partial^2 f}{\partial y^2} & \frac{\partial^2 f}{\partial y \partial z} & \frac{\partial^2 f}{\partial y \partial w} \\
\frac{\partial^2 f}{\partial z \partial x} & \frac{\partial^2 f}{\partial z \partial y} & \frac{\partial^2 f}{\partial z^2} & \frac{\partial^2 f}{\partial z \partial w} \\
\frac{\partial^2 f}{\partial w \partial x} & \frac{\partial^2 f}{\partial w \partial y} & \frac{\partial^2 f}{\partial w \partial z} & \frac{\partial^2 f}{\partial w^2}
\end{bmatrix}.
\end{equation}

Substituint les derivades:
\begin{equation}
H = \begin{bmatrix}
12x y^4 z w^2 + 6z^2 & 24x^2 y^3 z w^2 & 6x^2 y^4 w^2 + 12x z & 12x^2 y^4 z w \\
24x^2 y^3 z w^2 & 24x^3 y^2 z w^2 – 12y w^4 & 8x^3 y^3 w^2 & 16x^3 y^3 z w – 24y^2 w^3 \\
6x^2 y^4 w^2 + 12x z & 8x^3 y^3 w^2 & 6x^2 & 4x^3 y^4 w \\
12x^2 y^4 z w & 16x^3 y^3 z w – 24y^2 w^3 & 4x^3 y^4 w & 4x^3 y^4 z – 24y^3 w^2
\end{bmatrix}.
\end{equation}

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Pas 4: Avaluació en el punt $(2, -1, 1, -1)$

Avaluem cada entrada en $x = 2$, $y = -1$, $z = 1$, $w = -1$:

1. $\frac{\partial^2 f}{\partial x^2} = 12x y^4 z w^2 + 6z^2$:
   $$y^4 = (-1)^4 = 1, \quad z = 1, \quad w^2 = (-1)^2 = 1, \quad z^2 = 1^2 = 1,$$
   $$12 \cdot 2 \cdot 1 \cdot 1 \cdot 1 + 6 \cdot 1 = 24 + 6 = 30.$$
2. $\frac{\partial^2 f}{\partial x \partial y} = 24x^2 y^3 z w^2$:
   $$x^2 = 2^2 = 4, \quad y^3 = (-1)^3 = -1, \quad z = 1, \quad w^2 = 1,$$
   $$24 \cdot 4 \cdot (-1) \cdot 1 \cdot 1 = -96.$$
3. $\frac{\partial^2 f}{\partial x \partial z} = 6x^2 y^4 w^2 + 12x z$:
   $$x^2 = 4, \quad y^4 = 1, \quad w^2 = 1, \quad x = 2, \quad z = 1,$$
   $$6 \cdot 4 \cdot 1 \cdot 1 + 12 \cdot 2 \cdot 1 = 24 + 24 = 48.$$
4. $\frac{\partial^2 f}{\partial x \partial w} = 12x^2 y^4 z w$:
   $$x^2 = 4, \quad y^4 = 1, \quad z = 1, \quad w = -1,$$
   $$12 \cdot 4 \cdot 1 \cdot 1 \cdot (-1) = -48.$$
5. $\frac{\partial^2 f}{\partial y \partial x} = 24x^2 y^3 z w^2 = -96$ (igual que $\frac{\partial^2 f}{\partial x \partial y}$).
6. $\frac{\partial^2 f}{\partial y^2} = 24x^3 y^2 z w^2 – 12y w^4$:
   $$x^3 = 2^3 = 8, \quad y^2 = (-1)^2 = 1, \quad z = 1, \quad w^2 = 1, \quad y = -1, \quad w^4 = (-1)^4 = 1,$$
   $$24 \cdot 8 \cdot 1 \cdot 1 \cdot 1 – 12 \cdot (-1) \cdot 1 = 192 + 12 = 204.$$
7. $\frac{\partial^2 f}{\partial y \partial z} = 8x^3 y^3 w^2$:
   $$x^3 = 8, \quad y^3 = -1, \quad w^2 = 1,$$
   $$8 \cdot 8 \cdot (-1) \cdot 1 = -64.$$
8. $\frac{\partial^2 f}{\partial y \partial w} = 16x^3 y^3 z w – 24y^2 w^3$:
   $$x^3 = 8, \quad y^3 = -1, \quad z = 1, \quad w = -1, \quad y^2 = 1, \quad w^3 = (-1)^3 = -1,$$
   $$16 \cdot 8 \cdot (-1) \cdot 1 \cdot (-1) – 24 \cdot 1 \cdot (-1) = 128 + 24 = 152.$$
9. $\frac{\partial^2 f}{\partial z \partial x} = 6x^2 y^4 w^2 + 12x z = 48$ (igual que $\frac{\partial^2 f}{\partial x \partial z}$).
10. $\frac{\partial^2 f}{\partial z \partial y} = 8x^3 y^3 w^2 = -64$ (igual que $\frac{\partial^2 f}{\partial y \partial z}$).
11. (\frac{\partial^2 f}{\partial z^2} = 6x^2):
    $$x^2 = 4,$$
    $$6 \cdot 4 = 24.$$
12. $\frac{\partial^2 f}{\partial z \partial w} = 4x^3 y^4 w$:
    $$x^3 = 8, \quad y^4 = 1, \quad w = -1,$$
    $$4 \cdot 8 \cdot 1 \cdot (-1) = -32.$$
13. $\frac{\partial^2 f}{\partial w \partial x} = 12x^2 y^4 z w = -48$ (igual que $\frac{\partial^2 f}{\partial x \partial w}$).
14. $\frac{\partial^2 f}{\partial w \partial y} = 16x^3 y^3 z w – 24y^2 w^3 = 152$ (igual que $\frac{\partial^2 f}{\partial y \partial w}$).
15. $\frac{\partial^2 f}{\partial w \partial z} = 4x^3 y^4 w = -32$ (igual que $\frac{\partial^2 f}{\partial z \partial w}$).
16. $\frac{\partial^2 f}{\partial w^2} = 4x^3 y^4 z – 24y^3 w^2$:
    $$x^3 = 8, \quad y^4 = 1, \quad z = 1, \quad y^3 = -1, \quad w^2 = 1,$$
    $$4 \cdot 8 \cdot 1 \cdot 1 – 24 \cdot (-1) \cdot 1 = 32 + 24 = 56.$$

La matriu Hessiana en $(2, -1, 1, -1)$ és:
\begin{equation}
H = \begin{bmatrix}
30 & -96 & 48 & -48 \\
-96 & 204 & -64 & 152 \\
48 & -64 & 24 & -32 \\
-48 & 152 & -32 & 56
\end{bmatrix}.
\end{equation}

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Resposta final

La matriu Hessiana de la funció $f(x, y, z, w) = 2x^3 y^4 z w^2 – 2y^3 w^4 + 3x^2 z^2$ en el punt $(2, -1, 1, -1)$ és:
\begin{equation}
\boxed{\begin{bmatrix}
30 & -96 & 48 & -48 \\
-96 & 204 & -64 & 152 \\
48 & -64 & 24 & -32 \\
-48 & 152 & -32 & 56
\end{bmatrix}}
\end{equation}