Concave function second derivative is negative then it is strictly concave, but the opposite is not true, as shown by . 4. If is concave and differentiable, then it is bounded above by its first-order Taylor approximation: 5. A continuous function on is concave if and only if for any and in 6. If a function is concave, and , then is subadditive on formula_21. Proof: 1. A function is concave over a convex set if and only if the function is a convex function over the set. 2. The sum of two concave functions is itself concave and so is the
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