Question: What happens to concavity when functions are added?

1. If f(x) and g(x) are concave for all x, is f(x) + g(x) concave up for all x?
2. If f(x) is concave up for all x and g(x) is concave down for all x, what can you say about the concavity of f(x) + g(x)? For example, what happens if f(x) and g(x) are both polynomials of degree2?
3. If f(x) is concave up for all x and g(x) is concave down for all x, is it possible for f(x) + g(x) to change concavity infinitely often?

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