The function f ∗ ∗ is called the convex conjugate function of f . For historical reasons (rooted in analytic mechanics), the conjugate variable is often denoted p , instead of x ∗ ∗ . If the convex function f is defined on the whole line and is everywhere differentiable, then f ∗ ∗ ( p ) = sup x ∈ ∈ I ( p x − − f ( x ) ) = ( p x − − f ( x ) ) | x = ( f ′ ) − − 1 ( p ) can be interpreted as the negative of the y -intercept of the tangent line to the graph of f that has slope p .