Concavity of Differentiable Functions
Contents |
Definition
Let $f$ be a differentiable real function on some interval $\mathbb I$.
Upward Concavity
The graph of $f$ is said to be concave upward on $\mathbb I$ iff:
- $\forall x_1,x_2 \in \mathbb I : x_1 > x_2 \implies f'(x_1) > f'(x_2)$
This is (for differentiable functions) equivalent to $f$ being convex.
Proof of this claim is given in Derivative of Convex or Concave Function.
Downward Concavity
The graph of $f$ is said to be concave downward on $\mathbb I$ iff:
- $\forall x_1,x_2 \in \mathbb I : x_1 > x_2 \implies f'(x_1) < f'(x_2)$
This is (for differentiable functions) equivalent to $f$ being concave.
Proof of this claim is given in Derivative of Convex or Concave Function.
According to the strict house style, above definitions should be on separate pages. Page needs to be moved to appropriate namespace under an appropriate name. -Lord_Farin
Also, there is a bit of inconsistency between the newer pages and the old ones regarding the use of "increasing" vs "strictly increasing". In general, I did a poor job of blending the new concavity pages into the old concave/convex ones because I didn't register that they were the same concept until after the fact. - GFauxPas
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Sources
- Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus: 8th Edition (2005): $\S 3.3, \S3.4$
- For a video presentation of the contents of this page, visit the Khan Academy.
