In the book http://www.athenasc.com/nonlinbook.html by Dimitri P. Bertsekas, there is a proposition A.23 in Appendix A that :
Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be twice continuously differentiable over an open sphere $S$ centered at vector $x$.
For all $y$ such that $x+y \in S$, there exists and $\alpha \in [0,1]$ such that
$$f(x+y)=f(x)+y^T\nabla f(x) + \frac{1}{2}y^T\nabla^2 f(x+\alpha y)y$$
I am not able to prove it. I suspect it is somewhere related to taylor series and convexity, but I am not sure how to prove it.
Consider the function $g(t)=f(x+t y)$ where $t$ is in $[0,1]$. Apply the Taylor's expansion with the Lagrange reminder.
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