Final answer:
The question is about whether the product of two convex, nondecreasing (or nonincreasing), positive functions is convex. The product of two convex functions is not necessarily convex unless additional conditions are met. Thus, the statement cannot be definitively answered as true without further information.
Step-by-step explanation:
The question asks if the product of two convex functions, which are also either nondecreasing or nonincreasing and positive over an interval, is also convex. Without additional context, we cannot guarantee that the product of two convex functions will also be convex. For two functions f and g, being convex means that for any two points x, y in the domain and any λ where 0 ≤ λ ≤ 1, the following inequality holds:
f(λx + (1-λ)y) ≤ λf(x) + (1-λ)f(y)
Similarly for g:
g(λx + (1-λ)y) ≤ λg(x) + (1-λ)g(y)
To show that fg is convex, we would need to show that:
(fg)(λx + (1-λ)y) ≤ λ(fg)(x) + (1-λ)(fg)(y)
For general convex functions, this is not necessarily true. The functions being positive, nondecreasing, or nonincreasing are helpful properties, but they do not provide enough information on their own to prove the convexity of the product. Therefore, the statement is not conclusively correct without additional properties or constraints on f and g.