Prove: Increasing Function f(x) Has a Positive Value at a

[Ki-nana18]

Homework Statement

Suppose that a function f(x) is increasing and concave up. Show that there is a number "a", such that f(a)>0.

The Attempt at a Solution

How would I go about showing this?

 

[CompuChip]

What do "increasing" and "concave up" mean (in terms of derivatives)?

I think the best way to start the proof is: Pick an arbitrary x₀ in the domain.
Then either f(x₀) > 0 and you are done, or f(x₀) <= 0. Now probably your intuition tells you that somewhere to the right of x₀, f must become positive, right?
You can formalise that by taking some x > x₀, and making estimates for f(x) in terms of f(x₀) and positive quantities, until you find an x = a such that f(a) > 0.