gen-define-a-convex-function-and-give-one-short-worked-example-7d70

Define a convex function and give one short worked example.

# Convex Function ## Definition A function f: ℝⁿ → ℝ is **convex** if for all points **x**, **y** in its domain and all λ ∈ [0,1]: **f(λx + (1-λ)y) ≤ λf(x) + (1-λ)f(y)** Geometrically: the line segment connecting any two points on the graph lies above (or on) the graph itself. ## Worked Example **Show that f(x) = x² is convex.** Take any two points x, y ∈ ℝ and λ ∈ [0,1]. We need to verify: f(λx + (1-λ)y) ≤ λf(x) + (1-λ)f(y) **Left side:** f(λx + (1-λ)y) = (λx + (1-λ)y)² = λ²x² + 2λ(1-λ)xy + (1-λ)²y² **Right side:** λf(x) + (1-λ)f(y) = λx² + (1-λ)y² **Check the inequality:** (Right - Left) = λx² + (1-λ)y² - λ²x² - 2λ(1-λ)xy - (1-λ)²y² = λ(1-λ)x² + λ(1-λ)y² - 2λ(1-λ)xy = λ(1-λ)(x² - 2xy + y²) = λ(1-λ)(x - y)² ≥ 0 ✓ Since the difference is non-negative, **f(x) = x² is convex**.