# Bound a $$\mathcal{C}^2$$ function!

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.

Note by Austin Stromme
4 years ago

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Hint #1: Taylor Series.

- 4 years ago