Convex, concave, and related functions arise naturally in economics. They include the indirect utility function, cost function, expenditure function, and profit function.

Once the incident rays strike the mirror, reflect them according to the two rules of reflection for concave mirrors, which you verified in Part A. The ray that passes through the focal point on the way to the …

5.6 Determining Concavity Notes Write your questions and thoughts here! What is concavity? The state or quality of being concave. Concave up: Concave down: If a function is concave up (like a …

However, a linear function is neither strictly concave nor strictly convex, as the inequality is never strict.

The slope of the tangent line getting bigger implies that the derivative is getting bigger (increasing), so f0(x) is increasing when f(x) is concave up. We know, from last time, that the sign of the derivative …

Maxima for differentiable concave functions From elementary calculus we know that f : < ! < and if f 0(x) exists and x is in the interior of its domain, then a necessary condition for x to be a local maximum of …

Proposition 1: The sum of concave functions is concave 1 See Module 4 for a proof of the equivalence of D1 and D2.