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) object off the optical axis of the lens is imaged, where rays pass through the lens at an angle to the axis θ. Rays that pass through the centre of a lens of focal length "f" are focused at a point with distance from the axis. Rays passing through the outer margins of the lens are focused at different points, either further from the axis (positive coma) or closer to the axis (negative coma). In general, a bundle of parallel rays passing through the lens at a fixed distance from the centre of the lens are focused to a ring-shaped