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- Homework Statement
- What is the maximum (or minimum) of a multi-variable Taylor Series?

- Relevant Equations
- Assume that the function ##f## to be expanded has three variables ##x, y, z##, so its Taylor series expansion is (centered at ##\left(a, b, c\right)##),

\begin{align*}

f\left(x, y, z\right) &= f\left(a, b, c\right) + \left[\frac{\partial f\left(a, b, c\right)}{\partial x}\left(x - a\right) + \frac{\partial f\left(a, b, c\right)}{\partial y}\left(y - b\right) + \frac{\partial f\left(a, b, c\right)}{\partial z}\left(z - c\right)\right] + \frac{1}{2}\left[\frac{\partial^2 f\left(a, b, c\right)}{\partial x^2}\left(x - a\right)^2 + \frac{\partial^2 f\left(a, b, c\right)}{\partial x \partial y}\left(x - a\right)\left(y - b\right) + \frac{\partial^2 f\left(a, b, c\right)}{\partial x \partial z}\left(x - a\right)\left(z - c\right) + \frac{\partial^2 f\left(a, b, c\right)}{\partial y \partial x}\left(y - b\right)\left(x - a\right) + \frac{\partial^2 f\left(a, b, c\right)}{\partial y^2}\left(y - b\right)^2 + \frac{\partial^2 f\left(a, b, c\right)}{\partial y \partial z}\left(y - b\right)\left(z - c\right) + \frac{\partial^2 f\left(a, b, c\right)}{\partial z \partial x}\left(z - c\right)\left(x - a\right) + \frac{\partial^2 f\left(a, b, c\right)}{\partial z \partial y}\left(z - c\right)\left(y - b\right) + \frac{\partial^2 f\left(a, b, c\right)}{\partial z^2}\left(z - c\right)^2\right],

\end{align*}

which is up to the second-order term.

Firstly, the matrix notation of the series is,

\begin{align*}

f\left(x, y, z\right) &= f\left(a, b, c\right) + \left(\mathbf{x} - \mathbf{a}\right)^T \frac{\partial f\left(a, b, c\right)}{\partial \mathbf{x}} + \frac{1}{2}\left(\mathbf{x} - \mathbf{a}\right)^T \frac{\partial^2 f\left(a, b, c\right)}{\partial \mathbf{x}^2} \left(\mathbf{x} - \mathbf{a}\right),

\end{align*}

where ##\mathbf{x} = \left[x, y, z\right]^T## and ##\mathbf{a} = \left[a, b, c\right]^T##. The first derivative of the series is set to zero to find the maximum or minimum point, then how to apply the first derivative? Is the first derivative matrix to be applied is ##\frac{\partial}{\partial \mathbf{x}} = \left[\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right]^T##?

Thank you in advance.

\begin{align*}

f\left(x, y, z\right) &= f\left(a, b, c\right) + \left(\mathbf{x} - \mathbf{a}\right)^T \frac{\partial f\left(a, b, c\right)}{\partial \mathbf{x}} + \frac{1}{2}\left(\mathbf{x} - \mathbf{a}\right)^T \frac{\partial^2 f\left(a, b, c\right)}{\partial \mathbf{x}^2} \left(\mathbf{x} - \mathbf{a}\right),

\end{align*}

where ##\mathbf{x} = \left[x, y, z\right]^T## and ##\mathbf{a} = \left[a, b, c\right]^T##. The first derivative of the series is set to zero to find the maximum or minimum point, then how to apply the first derivative? Is the first derivative matrix to be applied is ##\frac{\partial}{\partial \mathbf{x}} = \left[\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right]^T##?

Thank you in advance.