```markdown
## Newton Step

### Formula
\[
\Delta x_{nt} = -\nabla^2 f(x)^{-1} \nabla f(x)
\]

### Interpretations
- **\(x + \Delta x_{nt}\) minimizes second order approximation**:
  \[
  \hat{f}(x + v) = f(x) + \nabla f(x)^T v + \frac{1}{2} v^T \nabla^2 f(x) v
  \]

- **\(x + \Delta x_{nt}\) solves linearized optimality condition**:
  \[
  \nabla f(x + v) \approx \nabla \hat{f}(x + v) = \nabla f(x) + \nabla^2 f(x) v = 0
  \]

### Diagram
- The diagram illustrates the concept of Newton's method:
  - A curve \(f\) with a point \((x, f(x))\).
  - A tangent line \(f'\) at \((x, f'(x))\).
  - A second point \((x + \Delta x_{nt}, f(x + \Delta x_{nt}))\) on the curve.
  - A tangent line \(f'^{\prime}\) at \((x + \Delta x_{nt}, f'(x + \Delta x_{nt}))\).

*Source: A. d'Aspremont, Convex Optimization M2.*
```
