TITLE: Taylor theorem equation
QUESTION [6 upvotes]: I have one question about Taylor theorem.
Originally, Taylor theorem is represented as
$$f(x) = f(a) + f'(a)(x-a) + \ldots$$
But my book says
Suppose that $$f : \mathbb{R}^n \to \mathbb{R}$$ is continuously differentiable and that $p\in\mathbb{R}^n$
Then we have that
$$f(x + p) = f(x) + ∇f(x+tp)^Tp$$
But I don't understand how to derive this equation from Taylor theorem.

REPLY [7 votes]: This equality follows from the mean value theorem applied to the function $ g (s) = f (x + sp) , 0 \leq s \leq 1$.
By the mean value theorem, there exists $ t \in (0,1)$ such that $ g (1) -g (0) = g'(t) = \nabla f (x + tp)^T p$.