If a function is smooth (differentiable), we can use its derivatives to prove it is convex. A differentiable function over a convex set is convex if and only if for all x, y ∈ K, the condition is satisfied:
Step-by-step breakdown: The right side is the first-order Taylor expansion of f around x (the tangent line). This guarantees the actual function value f(y) always sits strictly above (or on) its tangent hyperplane.
Sets where the tangent hyperplane touches the curve.
The point we are comparing against the tangent.