i am looking to see if the concavity of h(x) is the same for both h(x) and f(x) if h(x) is proportional to f(x) by a non negative factor.
Let h(x)=x^2 * f(x) , and f(x) is concave up everywhere. A graph of f is shown, though the book doesnt give the graph a name, f(x) is seen to be f(x)=x(x-4). The book asks the question is h(x) concave up at x=-0.5?
i know the answer to be yes, however instead of finding a graph of h''(x) i said that since h(x) is a function whose behavior is proportional to f(x) by a non-negative factor, where f(x) is concave up, h(x) is also concave up.
So the question is, was it only by chance i was right? I only used the given function as an example, but i need a general theory on the matter