Second-Order Multivariable Taylor Series

Theorem

If $f:U\subset {\mathbb{R}}^n\to \mathbb{R}$ is $C^3$ then: $f(x_0+h)=f(x_0)+{\sum }_{i=1}^n{h}_i\frac{\partial f}{\partial x_i}(x_0)+\frac{1}{2}{\sum }_{i,j=1}^n{h}_i{h}_j\frac{{\partial }^{2}f}{\partial x_i\partial x_j}(x_0)+R_2(x_0,h)$ Note that ${\lim }_{||h||\to 0}\frac{R_2(x_0,h)}{||h|{|}^2}=0$ The second-order taylor approxmiation is: $f(x_0+h)\sim f(x_0)+[Df(x_0)]h+Hf(h)$

Generalization for Function at Point

$$f(x)\sim f(a)+Df(a)(x-a)+\frac{1}{2}(x-a{)}^{T}Hf(a)(x-a)$$