Is $f$ always positive if $f'>0$, $f''<0$?

Question

Is functions $f$ always positive if $f'>0$ and $f''<0$?

I intuitively can think that this is true but I want more details about the reason why this satisfies or not.

Thank you.

Answer 1

What about the counterexample $f(x)= - \frac{1}{x}$ defined on $(0, \infty)$?

Answer 2

If you take $f(x) = - e^{-x} + 2$ it has both positive and negative values and $f’(x) > 0$ and $f’’(x)< 0$