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
Lens (optics) real object at the location of that virtual image. Likewise, it appears to a subsequent lens as if it were an object at that location, so that second lens could again focus that light into a real image, "S" then being measured from the virtual image location behind the first lens to the second lens. This is exactly what the eye does when looking through a magnifying glass. The magnifying glass creates a (magnified) virtual image behind the magnifying glass, but those rays are then re-imaged by the lens of the eye to create a "real image" on the retina.