The definition of the **concavity** of graphs is introduced along with inflection points. Examples, with detailed solutions, are used to clarify the concept of concavity.

**Example 1:** Let us consider the graph below. Note that the slope of the tangent line (first
derivative) increases. The graph in the figure below is called **concave up**.

Figure 1
**Example 2:** The slope of the tangent line (first derivative) decreases in the graph below. We call the graph below **concave down**.

Figure 2

## Definition of Concavity

Let f ' be the first derivative of function f that is differentiable on a given interval I, the graph of f is

(i) **concave up** on the interval I, if f ' is increasing on I

, or

(ii) **concave down** on the interval I, if f ' is decreasing on I.
The sign of the second derivative informs us when is f ' increasing or decreasing.

## Theorem

Let f '' be the second derivative of function f on a given interval I, the graph of f is

(i) **concave up** on I if f ''(x) > 0 on the interval I.

(ii) **concave down** on I if f ''(x) < 0 on the interval I.
**Example 3:** Determine the values of parameter a for which the graph of function f, defined below, is concave up or down.

*f(x) = a x*^{ 2} + b x + c

**Solution to Example 3:**