Concavity of a function

Definition

The concavity of a function, or more precisely the sense of concavity of a function, describes the way the derivative of the function is changing. There are two determinate senses of concavity: concave up and concave down. Note that it is possible for a function to be neither concave up nor concave down.

Definition in terms of convex combinations

A function $f$ defined on an interval $I$ is termed concave up if it satisfies the following condition: for any $x_{1},x_{2}\in I$ with $x_{1}<x_{2}$, and any $t\in (0,1)$, we have:

$f(tx_{1}+(1-t)x_{2})<tf(x_{1})+(1-t)f(x_{2})$

Geometrically, what this means is that the chord joining the points $(x_{1},f(x_{1}))$ and $(x_{2},f(x_{2}))$ in the graph of $f$ lies above the part of the graph of $f$ between $x_{1}$ and $x_{2}$.

A function $f$ defined on an interval $I$ is termed concave down if it satisfies the following condition: for any $x_{1},x_{2}\in I$ with $x_{1}<x_{2}$, and any $t\in (0,1)$, we have:

$f(tx_{1}+(1-t)x_{2})>tf(x_{1})+(1-t)f(x_{2})$

Geometrically, what this means is that the chord joining the points $(x_{1},f(x_{1}))$ and $(x_{2},f(x_{2}))$ in the graph of $f$ lies below the part of the graph of $f$ between $x_{1}$ and $x_{2}$.

Definition in terms of first derivative

A function $f$ defined on an interval $I$ is termed concave up if it satisfies the following conditions:

1. The derivative of $f$ is defined everywhere on $I$ except at possibly a discrete set of points.
2. Consider each of the intervals where the derivative of $f$ is defined. The derivative of $f$ is an increasing function on that interval.
3. At each of the points in the interior of the domain where the derivative of $f$ is not defined, the following are true: $f$ is continuous, $f$ is left differentiable, $f$ is right differentiable, and the left hand derivative of $f$ is less than the right hand derivative of $f$.
4. At the endpoints of the interval, if they exist, $f$ is appropriately one-sided continuous and one-sided differentiable.

A function $f$ defined on an interval $I$ is termed concave down if it satisfies the following conditions:

1. The derivative of $f$ is defined everywhere on $I$ except at possibly a discrete set of points.
2. Consider each of the intervals where the derivative of $f$ is defined. The derivative of $f$ is a decreasing function on that interval.
3. At each of the points in the interior of the domain where the derivative of $f$ is not defined, the following are true: $f$ is continuous, $f$ is left differentiable, $f$ is right differentiable, and the left hand derivative of $f$ is greater than the right hand derivative of $f$.
4. At the endpoints of the interval, if they exist, $f$ is appropriately one-sided continuous and one-sided differentiable.