# Concave function

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## Contents

## Definition

### For a function of a single variable

Suppose is a function from an interval in the real numbers, to the real numbers. We say that is a **concave function** on if:

.

We say that is a **strictly concave function** on if:

.

Concave functions could be of many type:

- Concave functions that are increasing throughout, and in the limit still have positive first derivative.
- Concave functions that are increasing throughout, and in the right limit have zero derivative.
- Concave functions that start off increasing and end up decreasing.
- Concave functions that start off with zero derivative and end up decreasing.
- Concave functions that start off with negative first derivative.

## Facts

### Negative of convex

A function is concave if and only if the function is a convex function. Similarly, is a strictly concave function if and only if is a strictly convex function.

### Closed under addition

A linear combination of concave functions with *positive* coefficients is again concave.

### Inverse function of a concave function is concave

If is a concave function, and it is invertible, then the inverse function is also a concave function.