Let \(f\colon \mathbb{R}^2 \to \mathbb{R}\) be a \(\mathcal{C}^2\) function. Suppose that \(M>0\) is a real number such that

\(|f_{xx} | \leq M \), \(|f_{xy}| \leq M \), and \(|f_{yy}| \leq M\).

Show that

\( | (f(\mathbf{x}+\mathbf{h}) - f(\mathbf{x})) -\nabla f(\mathbf{x}) \cdot \mathbf{h}| \leq M \| \mathbf{h} \|^2. \)

Look below for hints.

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