There is not such function in $\mathbb{R}$ such that $f>0$, $f'>0$ and $f''<0$. Prove that there isn't a function $f \colon \mathbb{R} \to \mathbb{R}$  such that
$f&gt;0$, $f'&gt;0$ and $f''&lt;0$.

Any suggestion will be appreciated.
Since $f$ is strictly concave and non-constant, either $\lim_{x \to -\infty} f(x)=-\infty$ or $\lim_{x \to +\infty} f(x)=-\infty$. In both cases, $f$ can't be always positive.
$f(x)= \sqrt{x}+1$ is perhaps a counter example. It is always positive, has a positive derivative and concave down.