sum of concave functions?

Question

how do you show that the sum of two concave functions is also concave?

Answer 1

While gudspeling's method is a quick way to verify this property for most functions you'll actually encounter, it only works for a specific type of function (i.e. twice continuously differentiable). So it's not proving the _general_ case for a function of a single variable.

The more general definition of concavity that works for _any_ real valued function defined over some convex Euclidian subspace is as follows. The function f() is concave over its domain if for any two points in the domain, x and y:
f( ax + (1-a)y ) ≥ af(x)+(1-a)f(y),
where 0
Let h(x) = f(x) + g(x), where f() and g() are assumed to both be concave over domain D. Then:
h(ax + (1-a)y) = f(ax+(1-a)y) + g(ax+(1-a)y) <--- definition of h()
≥ af(x)+(1-a)f(y) + ag(x)+(1-a)g(y) <--- concavity of f() and g()
= a[f(x)+g(x)]+(1-a)[f(y) + g(y)] <--- rearranging.
= ah(x) + (1-a)h(y) <--- definition of h()

Thus h(ax + (1-a)y) ≥ ah(x) + (1-a)h(y), which means that h() is concave over D.

[Note that the derivative-based method also needs to be tweaked when the functions have multiple arguments. The proof given here works for both the single- and multiple-argument cases -- x and y could be either numbers or vectors.]

Answer 2

Concave Functions

Answer 3

You need to state whether the functions you are talking about are everywhere concave up or everywhere concave down. In otherwords, the functions are not concave down for part of their graph and concave up for the rest of the graph. Further you need to state whether the two functions have the same concavity ( both concave up or both concave down).

Answer 4

f(x) = g(x) + h(x)

A function (F(x)) is concave if F"(x) > 0

g(x) and h(x) are concave functions:
g"(x) > 0
h"(x) > 0
f"(x) = g"(x) + h"(x) > 0
f"(x) > 0

f(x) is concave.