gen-briefly-explain-a-convex-function-and-one-common-misconcepti-470d

Briefly explain a convex function and one common misconception about it.

## Convex Function A **convex function** is one where the line segment connecting any two points on the graph lies *above or on* the graph itself. Mathematically, for any two points x₁ and x₂ and any λ ∈ [0,1]: f(λx₁ + (1-λ)x₂) ≤ λf(x₁) + (1-λ)f(x₂) Visually, convex functions "curve upward" or bowl-shaped (like f(x) = x²). ## Common Misconception **Misconception:** *A convex function must be differentiable or smooth.* **Reality:** Convex functions can have sharp corners or kinks. The classic example is f(x) = |x|, which is convex but not differentiable at x = 0. Convexity is about the shape of the function, not its smoothness.