To see this, first note that if f as a convex function on the real line is differentiable and x ¯ ¯ is a critical point of the function of x ↦ ↦ p ⋅ ⋅ x − − f ( x ) , then the supremum is achieved at x ¯ ¯ (by convexity, see the first figure in this Wikipedia page). Therefore, the Legendre transform of f is f ∗ ∗ ( p ) = p ⋅ ⋅ x ¯ ¯ − − f ( x ¯ ¯ ) .