I am following a proof that applies Taylor’s theorem on this document (http://www.gautampendse.com/software/lasso/webpage/pendseLassoShooting.pdf)
I am not understanding one of the terms explained on the formula (5.1):
Proposition 5.3 (positive semidefinite Hessian implies Convexity). Suppose $x$ is a $p\times1$ vector and $f(x)$ is a scalar function of $p$ variables with continuous second order derivatives defined on a convex domain $D$. If the Hessian $\nabla^2f(x)$ is positive semidefinite for all $x\in D$, then $f$ is convex.
Proof. By Taylor’s theorem for all $x$, $x+h\in D$ we can write: $$f(x + h) = f(x) + \nabla f(x)^Th + \frac{1}{2}h^T\nabla^2f(x + \theta h)h$$ for some $\theta\in(0,1)$.
I don’t understand why the term corresponding to the second derivative is: $$ \frac{1}{2}h^T\nabla^2f(x + \theta h)h,$$ and not: $$\frac{1}{2}h^T\nabla^2f(x)h$$
I understand that by $\theta\in(0,1)$, if $\theta=0$ then the term becomes the latter, but I don’t understand what happens when $\theta>0$.
This formula uses the Cauchy form of the remainder. $x + \theta h $ is a number between $x$ and $x+h$. http://mathworld.wolfram.com/CauchyRemainder.html
It comes from the mean value theorem.