Waste less time on Facebook — follow Brilliant.
×

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
2 years, 10 months ago

No vote yet
1 vote

Comments

Sort by:

Top Newest

Hint #1: Taylor Series. Austin Stromme · 2 years, 10 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...