Real image (such as the images on the right), real rays of light are always represented by full, solid lines; perceived or extrapolated rays of light are represented by dashed lines. A real image occurs where rays converge, whereas a virtual image occurs where rays only "appear" to converge. Real images can be produced by concave mirrors and converging lenses, only if the object is placed further away from the mirror/lens than the focal point and this real image is inverted. As the object approaches the focal point the image approaches infinity, and when the object passes the focal point the image
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