Taylors Theorem

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Suppose that $f:{\mathbb{R}}^n\to \mathbb{R}$ is continously differentiable and that $p\in {\mathbb{R}}^n$. Then we have

$$f(x+p)=f(x)+\nabla f(x+tp{)}^{T}p$$

For some $t\in (0,1)$.

Moreover, if twice differentiable, then we have

$$\nabla f(x+p)=\nabla f(x)+{\int }_0^1{\nabla }^{2}f(x+tp)pdt$$

Which combined gives

Formula

f(x+p) = f(x) + \nabla f(x)^Tp + \frac{1}{2}p^T\nabla^{2}f(x+tp)p

For some $t \in (0,1)$.

Note

This is often called the mean value theorem