Second Derivative of Concave Downward Function
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Contents |
Theorem
Let $f$ be a differentiable real function on some interval $\mathbb I$. If $f''\left(x\right) < 0$ for all $x$ on the interval, $f$ is concave down.
Proof
By Derivative of Monotone Function, if $\forall x \in \mathbb I : f''\left(x\right) < 0$ then $f'$ is strictly decreasing.
The result follows from the definition of downward-concavity for differentiable functions.
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Also see
Sources
- For a video presentation of the contents of this page, visit the Khan Academy.
