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.

#TaylorSeries #Analysis

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

Hint #1: Taylor Series.

Austin Stromme - 6 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Bound a \( \mathcal{C}^2 \) function!

Comments