Question

From Problem, it follows that the sum of concave functions is concave. Is the sum of quasi-concave functions necessarily quasi-concave? Problem: Show that if f(x1, x2,…, xn) and g(x1, x2,…, xn) are convex functions on a convex set S, then h(x1, x2,…, xn) = f(x1, x2,…, xn) = g(x1, x2,…, xn) is a convex function on S.

Answer 1

The sum of quasi-concave functions is not necessarily quasi-concave, which is different from the behavior of convex functions, where their sum is always convex.

Step-by-step explanation:

The question asks about the properties of quasi-concave functions when they are combined, specifically whether the sum of quasi-concave functions is necessarily quasi-concave. This is a concept within the field of mathematical optimization and convex analysis. First, it is important to understand that a function is quasi-concave if its level sets are convex; however, unlike convex functions, the sum of two quasi-concave functions is not necessarily quasi-concave. While a similar property holds for convex functions, where the sum of convex functions is convex, quasi-concavity is not preserved under addition due to the less stringent restrictions on quasi-concave functions compared to convex functions.