Concave function the graph of formula_1 is above the straight line joining the points formula_15 and formula_16. A function formula_1 is quasiconcave if the upper contour sets of the function formula_18 are convex sets. 1. A differentiable function is (strictly) concave on an interval if and only if its derivative function is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope. 2. Points where concavity changes (between concave and convex) are inflection points. 3. If is twice-differentiable, then is concave if and only if is non-positive (or, informally, if the "acceleration" is non-positive). If its
Luneburg lens one side, and formula_14 only on the other: The parameter formula_19 is a constant for any given ray, but differs between rays passing at different distances from the centre of the lens. For rays passing through the centre, it is zero. In some special cases, such as for Maxwell's fish-eye, this first order equation can be further integrated to give a formula for formula_14 as a function or formula_13. In general it provides the relative rates of change of formula_14 and formula_13, which may be integrated numerically to follow the path of the ray through the lens. Luneburg lens A