# Category:Concave Real Functions

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This category contains results about **Concave Real Functions**.

Definitions specific to this category can be found in Definitions/Concave Real Functions.

$f$ is **concave on $I$** if and only if:

- $\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} \ge \alpha \map f x + \beta \map f y$

## Also see

## Subcategories

This category has only the following subcategory.

### E

## Pages in category "Concave Real Functions"

The following 18 pages are in this category, out of 18 total.

### C

### E

### I

- Inverse of Strictly Decreasing Concave Real Function is Concave
- Inverse of Strictly Decreasing Strictly Concave Real Function is Strictly Concave
- Inverse of Strictly Increasing Concave Real Function is Convex
- Inverse of Strictly Increasing Convex Real Function is Concave
- Inverse of Strictly Increasing Strictly Concave Real Function is Strictly Convex
- Inverse of Strictly Increasing Strictly Convex Real Function is Strictly Concave

### R

- Real Function both Convex and Concave is Linear
- Real Function is Concave iff Derivative is Decreasing
- Real Function is Concave iff its Negative is Convex
- Real Function is Strictly Concave iff Derivative is Strictly Decreasing
- Real Function with Strictly Negative Second Derivative is Strictly Concave