```markdown
## First-order Condition

- **Differentiability**: 
  - A function \( f \) is differentiable if its domain \( \text{dom } f \) is open and the gradient 
    \[
    \nabla f(x) = \left( \frac{\partial f(x)}{\partial x_1}, \frac{\partial f(x)}{\partial x_2}, \ldots, \frac{\partial f(x)}{\partial x_n} \right)
    \]
    exists at each \( x \in \text{dom } f \).

- **1st-order Condition**:
  - A differentiable function \( f \) with a convex domain is convex if and only if
    \[
    f(y) \geq f(x) + \nabla f(x)^T (y - x)
    \]
    for all \( x, y \in \text{dom } f \).

- **Illustration**:
  - The diagram shows a curve representing \( f(y) \) and a tangent line at point \((x, f(x))\).
  - The tangent line equation is \( f(x) + \nabla f(x)^T (y - x) \).
  - The first-order approximation of \( f \) is a global underestimator.

- **Source**:
  - A. d’Aspremont. Convex Optimization M2. Slide 9/67.
```
