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If $a<b$, what qualities of $f(x)$ are needed to be certain that $f(a) < f(b)$ ? For example, is it enough to say that $f(x)$ is monotonic and positive?

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Perhaps you are confusing monotonicity with the fact that if the derivative is positive, then the function is strictly increasing. – Srivatsan Sep 23 '11 at 16:41
up vote 2 down vote accepted

If you need this for all $a$ and $b$ then by definition $f$ is strictly increasing. Being positive is not relevant.

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