Why a decreasing, positive, concave function cannot exist

Text

Show that a function that is always decreasing, always positive and always concave down cannot be drawn on all of $\mathbb{R}$.

Solution

Let $f$ be defined on all of $\mathbb{R}$, decreasing and concave down, i.e. $f'<0$ and $f''<0$ everywhere. Since $f''<0$, the derivative $f'$ is itself decreasing: then for every $x$ greater than a fixed $x_0$ we have $f'(x) \le f'(x_0) = m < 0.$ The slope stays below a negative constant $m$: the function drops at least as fast as the line of slope $m$. As $x\to+\infty$ this forces $f(x) \le f(x_0) + m\,(x - x_0) \xrightarrow[x\to+\infty]{} -\infty,$ so eventually $f$ becomes negative, contradicting the assumption that it is always positive. Hence a decreasing, positive, concave-down function on all of $\mathbb{R}$ cannot exist: downward concavity prevents the curve from “flattening out” above the $x$-axis, as a convex function (e.g. $e^{-x}$) would instead do.