Basic Properties of Twice Differentiability at a Point

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the real numbers, and the Euclidean norm $\lVert\,\cdot\,\rVert$, dot product, distance $d_{E}$ and notion of openness, are as fixed there. Write $\mathcal{S}(n)$ for the set of symmetric real $n\times n$ matrices, $Bh$ for the matrix-vector product, $0_{n}$ for the real $n\times n$ matrix all of whose entries are $0$, which belongs to $\mathcal{S}(n)$, and $\preceq$ for the positive semidefinite ordering on $\mathcal{S}(n)$.

Let $U\subseteq\mathbb{R}^{n}$ be open and let $f:U\to\mathbb{R}$. Twice differentiability at a point is as defined there. Then the following hold.

1. (The first-order coefficient is the gradient) (i) Suppose $f$ is twice differentiable at $y\in U$ with first-order coefficient $p$ and Hessian $B$. Then $f$ is differentiable at $y$ with derivative matrix the real matrix with one row and $n$ columns whose entry in row $1$ and column $i$ is the $i$th coordinate of $p$. Consequently all $n$ partial derivatives of $f$ exist at $y$ and $p$ is the gradient $Df(y)$.

2. (Functions of class $C^{2}$) (ii) Suppose $f$ is of class $C^{2}$ on $U$. Then $f$ is twice differentiable at every $y\in U$, with first-order coefficient the gradient $Df(y)$ and Hessian the Hessian matrix $D^{2}f(y)$, which belongs to $\mathcal{S}(n)$ by claim 2 of Equality of Mixed Partial Derivatives, and Symmetry and Continuity of the Hessian. The two uses of the notation $D^{2}f(y)$ therefore agree.

3. (Second-order condition at a local maximum) (iii) Suppose $f$ is twice differentiable at $y\in U$ with first-order coefficient $p$ and Hessian $B$, and that $f$ has a local maximum at $y$ relative to $U$. Then

$$p=0\qquad\text{and}\qquad B\preceq 0_{n}.$$