Let $f: D\subseteq \Bbb R^n \to \Bbb R$ be a $C^2$ function. I'm trying to show that the absolute value of the error of the first order Taylor approximation of $f(\mathbf x+\mathbf h)$ is bounded from above by $\frac{n^2M}{2}\|\mathbf h\|^2$, where $M$ is the maximum (or supremum?) of $|\partial_{ij}f(\mathbf x + t^*\mathbf h)|$ over all $i,j\in\{1,\dots, n\}$ and $t^*\in [0, 1]$.
Here's all I've been able to get:
$$|E| = |\frac 12\sum_{i,j=1}^n\partial_{ij}f(\mathbf x+t^*\mathbf h)h_ih_j| \le \frac 12\sum_{i,j=1}^n|\partial_{ij}f(\mathbf x+t^*\mathbf h)h_ih_j| = \frac 12\sum_{i,j=1}^n|\partial_{ij}f(\mathbf x+t^*\mathbf h)||h_ih_j| \le \frac M2\sum_{i,j=1}^n|h_ih_j| = \frac M2\left[\|\mathbf h\|^2+2\sum_{i<j}^n|h_ih_j|\right]$$
Searching the internet I see everyone else has a different bound for this. So I wonder if this one is even true. $\frac{n^2M}{2}\|\mathbf h\|^2$ is the bound my book claims, though.
Hint $$\sum_{i=1}^n |h_i| \le \sum_{i=1}^n \|\mathbf h\| = \|\mathbf h\|\sum_{i=1}^n 1$$