Differentiable Bounded Concave Real Function is Constant
Theorem
Let $f$ be a real function which is:
- $(1): \quad$ Differentiable on $\R$
- $(2): \quad$ Bounded on $\R$
- $(3): \quad$ Concave on $\R$.
Then $f$ is constant.
Proof
Let $f$ be differentiable and bounded on $\R$.
Let $f$ be concave on $\R$.
Let $\xi \in \R$.
Aiming for a contradiction, suppose $f' \left({\xi}\right) > 0$.
Then by Mean Value of Concave Real Function it follows that:
- $f \left({x}\right) \le f \left({\xi}\right) + f' \left({\xi}\right) \left({x - \xi}\right) \to -\infty$ as $x \to +\infty$
and therefore is not bounded.
Similarly, suppose $f' \left({\xi}\right) < 0$.
Then by Mean Value of Concave Real Function it follows that:
- $f \left({x}\right) \le f \left({\xi}\right) + f' \left({\xi}\right) \left({x - \xi}\right) \to -\infty$ as $x \to -\infty$
and therefore is likewise not bounded.
Hence $f' \left({\xi}\right) = 0$.
From Zero Derivative implies Constant Function, it follows that $f$ is constant.
$\blacksquare$
