Real Function is Concave iff its Negative is Convex

Theorem
Let $f$ be a real function.

Let $I \subseteq \R$ be an interval of $\R$.

Then $f$ is concave on $I$ iff $-f$ is convex on $\R$.

Necessary Condition
Let $f$ be concave on $I$.

Let $\alpha, \beta \in \R_{>0}$ such that $\alpha + \beta = 1$.

Then:

and so $-f$ is convex by definition.

Sufficient Condition
Let $-f$ be convex on $I$.

Let $\alpha, \beta \in \R_{>0}$ such that $\alpha + \beta = 1$.

Then:

and so $f$ is concave by definition.