# Concavity and Point of Inflection of Graphs

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

## Example 1: Concavity Up

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: Concavity Down

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.

## Definition of Point of Inflection

A point P on the graph of y = f(x) is a point of inflection if f is continuous at P and the concavity of the graph changes at P. In view of the above theorem, there is a point of inflection whenever the second derivative changes sign.

## Example 3

Determine the values of the leading coefficient*a*for which the graph of function

*f(x) = a x*is concave up or down.

^{ 2}+ b x + c## Solution to Example 3

We first find the first and second derivatives of function f.f '(x) = 2 a x + b

f ''(x) = 2 a

We now study the sign of f ''(x) which is equal to 2 a. If a is positive, f ''(x) is positive in the interval (-∞ , + ∞). According to the theorem above, the graph of f will be concave up for positive values of a.

If a is negative, the graph of f will be concave down on the interval (-∞ , + ∞) since f ''(x) = 2 a is negative.

The graphs of two quadratic functions are shown below: y = 2 x^2 - 2 x - 1 whose graph is convcave up because its leading coefficient (a = 2) is positive and y = - x^2 + 3 x + 1 whose graph is convcave down because its leading coefficient (a = -1) is negative.

## Example 4

a) Find the intervals on which the graph of f(x) = x^{ 4}- 2x

^{ 3}+ x is concave up, concave down and the point(s) of inflection if any.

b) Use a graphing calculator to graph f and confirm your answers to part a).

## Solution to Example 4

Let us find the first two derivatives of function f.a)

f '(x) = 4 x

^{ 3}- 6

^{ 2}+ 1

f ''(x) = 12

^{ 2}- 12 x

Find the zeros f ''(x).

12

^{ 2}- 12 x = 0

12 x (x - 1) = 0

Two zeros

x = 0 and x = 1

Study sign of f ''

The two zeros split the set of real numbers into three intervals. Select a value for x in each of the three intervals and find the sign of f''

We now use the table of sign and the theorem above to conclude that

in the interval (-∞ , 0); f '' is positive and therefore the graph of f is concave up

in the interval (0 , 1); f '' is negative and therefore the graph of f is concave down

in the interval (1 , +∞); f '' is positive and therefore the graph of f is concave up

The second derivative f '' changes sign at x = 0 and x = 1 and therefore the graph of f has two inflection point: (0 , f(0)) and (1 , f(1))

b)

The graph of f (blue) and f '' (red) are shown below. It can easily be seen that whenever f '' is negative (its graph is below the x-axis), the graph of f is concave down and whenever f '' is positive (its graph is above the x-axis) the graph of f is concave up.

Point (0,0) is a point of inflection where the concavity changes from up to down as x increases (from left to right) and point(1,0) is also a point of inflection where the concavity changes from down to up as x increases (from left to right).

## Example 5

The graph of the second derivative f '' of function f is shown below. Find the intervals where f is concave up, concave down and the point(s) of inflection if any.## Solution to Example 5

According to the graph of f '', the sign of f '' is given over the following intervalsa)

On the interval (-∞ , 2), the graph of f '' is below the x axis and therefore f '' is negative, hence f is concave down on the interval (-∞ , 2)

On the interval (2 , +∞), the graph of f '' is above the x axis and therefore f '' is positive, hence f is concave up on the interval (2 , +∞)

At x = 2, the sign of f '' changes and therefore x = 2 is a point of inflection.

## Example 6

The graph of the first derivative f ' of function f is shown below. Find the intervals where the graph of f is concave up, concave down and the point(s) of inflection if any.## Solution to Example 6

We use the graph of the first derivative f ' to find the sign of the second derivative and deduce the concavity of the graph of fa)

On the interval (-∞ , -2), f ' decreases and therefore f '' is negative; the graph of f is concave down

On the interval (-2 , -1), f ' increases and therefore f '' is positive; the graph of f is concave up

On the interval (-1 , 1), f ' decreases and therefore f '' is negative; ; the graph of f is concave down

On the interval (1 , ∞), f ' increases and therefore f '' is positive; ; the graph of f is concave up

The concavity of the graph of f changes at x = -2, x = -1 and x = 1 and therefore these are all point of inflection.

### More References and links

DerivativeCalculus Tutorials and Problems