Convex function an interval if and only if its second derivative is non-negative there; this gives a practical test for convexity. Visually, a twice differentiable convex function "curves up", without any bends the other way (inflection points). If its second derivative is positive at all points then the function is strictly convex, but the converse does not hold. For example, the second derivative of "f"("x") = "x" is "f" ′′("x") = 12"x", which is zero for "x" = 0, but "x" is strictly convex. 6. If is a convex function of one real variable, and , then is superadditive on the positive
Crepuscular rays Crepuscular rays Crepuscular rays (more commonly known as sunbeams, sun rays, splintered light, or god rays), in atmospheric optics, are rays of sunlight that appear to radiate from the point in the sky where the Sun is located. Shining through openings in clouds (particularly stratocumulus) or between other objects, these columns of sunlit particulates are separated by darker cloud-shadowed volumes. Despite seeming to converge toward the light source, the rays are in fact near-parallel shafts of sunlight. Their apparent convergence is a perspective effect. This phenomenon mimics the way parallel railway lines or long hallways, for example, seem to seek