The concavity of a function describes whether or not a tangent line near a point will be above or below the function. The concavity of a function can be calculated by checking the second derivative of the function.

Concave Up

If the second derivative at a point is positive then the function is said to be concave up near that point.

When a function is concave up the tangent line near that point will lie below the graph.

Example

In this example the function is depicted in blue and is always concave up. The tangent line at is shown in red and is always below the graph near

Concave Down

If the second derivative at a point is negative the function is said to be concave down near that point.

When a function is concave down the tangent line near that point will lie above the graph.

Example

In this example the function is depicted in blue and is always concave down. The tangent line at is shown in red and is always above the graph near