Introduction

This topic explores how to determine the concavity of functions using derivatives. The essential knowledge tells us that the graph of a function is concave up (down) on an open interval if the function's derivative is increasing (decreasing) on that interval, the second derivative of a function provides information about the function and its graph, including intervals of upward or downward concavity, and the second derivative of a function may be used to locate points of inflection for the graph of the original function. Understanding concavity helps us analyze the shape and behavior of function graphs.

Understanding Concavity

Visual and Mathematical Definitions

Concave Up: A function is concave up on an interval if the graph curves upward like a cup (∪). The tangent lines lie below the graph.

Concave Down: A function is concave down on an interval if the graph curves downward like a cap (∩). The tangent lines lie above the graph.

Taken from Study.com

Mathematical Definition (using $f^{'}$ ):

Concave up when $f^{'}$ is increasing
Concave down when $f^{'}$ is decreasing

Example 1:

For $f (x) = x^{2}$ :

$f^{'} (x) = 2 x$ is increasing for all $x$ (since the slope of $f^{'}$ is positive).

Therefore, $f (x) = x^{2}$ is concave up on $(- \infty, \infty)$ .

Taken from Desmos

Using the Second Derivative to Determine Concavity

The Second Derivative Test for Concavity

Since concavity depends on whether $f^{'}$ is increasing or decreasing, we use the derivative of $f^{'}$ (which is $f^{''}$ ) to test:

If $f^{''} (x) > 0$ on an interval, then $f$ is concave up on that interval.

If $f^{''} (x) < 0$ on an interval, then $f$ is concave down on that interval.

This works because:

$f^{''} > 0$ means $f^{'}$ is increasing
$f^{''} < 0$ means $f^{'}$ is decreasing

Finding Intervals of Concavity

Procedure:

Find $f^{''} (x)$
Find where $f^{''} (x) = 0$ or $f^{''} (x)$ is undefined
Create a sign chart for $f^{''} (x)$
Determine the sign of $f^{''} (x)$ on each interval
Conclude about concavity

Example 2:

Determine where $f (x) = x^{3} - 6 x^{2} + 9 x$ is concave up and concave down.

Find $f^{'} (x)$ : $f^{'} (x) = 3 x^{2} - 12 x + 9$

Find $f^{''} (x)$ : $f^{''} (x) = 6 x - 12 = 6 (x - 2)$

Find where $f^{''} (x) = 0$ : $6 (x - 2) = 0$ , therefore $x = 2$

Test intervals:

For $x < 2$ (test $x = 0$ ): $f^{''} (0) = - 12 < 0$
For $x > 2$ (test $x = 3$ ): $f^{''} (3) = 6 > 0$

Conclusion:

Concave down on $(- \infty, 2)$
Concave up on $(2, \infty)$

Taken from Desmos

Points of Inflection

Definition and Identification

A point of inflection is a point where the concavity changes (from concave up to concave down, or vice versa).

Requirements for a point of inflection at $x = c$ :

$f^{''} (c) = 0$ or $f^{''} (c)$ is undefined
The concavity changes at $x = c$

Important: Just because $f^{''} (c) = 0$ does NOT guarantee a point of inflection. The concavity must actually change.

Example 3:

Find all points of inflection of $g (x) = x^{4} - 4 x^{3}$ .

Find $g^{''} (x)$ : $g^{'} (x) = 4 x^{3} - 12 x^{2}$ $g^{''} (x) = 12 x^{2} - 24 x = 12 x (x - 2)$

Potential inflection points where $g^{''} (x) = 0$ : $x = 0$ and $x = 2$

Test concavity:

For $x < 0$ : $g^{''} (- 1) = 12 (1) (3) > 0$ (concave up)
For $0 < x < 2$ : $g^{''} (1) = 12 (1) (- 1) < 0$ (concave down)
For $x > 2$ : $g^{''} (3) = 12 (3) (1) > 0$ (concave up)

At $x = 0$ : concavity changes from up to down → inflection point

At $x = 2$ : concavity changes from down to up → inflection point

Points of inflection: $(0, 0)$ and $(2, - 16)$

Taken from Desmos

When Points Are Not Inflection Points

Example 4:

Determine if $h (x) = x^{4}$ has an inflection point at $x = 0$ .

Find $h^{''} (x)$ : $h^{'} (x) = 4 x^{3}$ $h^{''} (x) = 12 x^{2}$

Set $h^{''} (x) = 0$ : $12 x^{2} = 0$ $x = 0$

Test concavity:

For $x < 0$ : $h^{''} (- 1) = 12 > 0$ (concave up)
For $x > 0$ : $h^{''} (1) = 12 > 0$ (concave up)

At $x = 0$ : concavity does not change (remains concave up).

Therefore, $x = 0$ is not a point of inflection, even though $h^{''} (0) = 0$ .

Taken from Desmos

Analyzing More Complex Functions

Functions with Multiple Inflection Points

Example 5:

Find intervals of concavity and inflection points for $f (x) = x^{4} - 6 x^{2} + 8$ .

Find $f^{''} (x)$ :

$f^{'} (x) = 4 x^{3} - 12 x$

$f^{''} (x) = 12 x^{2} - 12 = 12 (x^{2} - 1) = 12 (x - 1) (x + 1)$

Potential inflection points: $x = - 1$ and $x = 1$

Test concavity:

For $x < - 1$ : $f^{''} (- 2) = 12 (3) > 0$ (concave up)
For $- 1 < x < 1$ : $f^{''} (0) = - 12 < 0$ (concave down)
For $x > 1$ : $f^{''} (2) = 12 (3) > 0$ (concave up)

Concavity:

Concave up on $(- \infty, - 1)$ and $(1, \infty)$
Concave down on $(- 1, 1)$

Inflection points: $(- 1, 3)$ and $(1, 3)$

Taken from Desmos

Trigonometric Functions

Example 6:

Determine the concavity of $f (x) = sin (x)$ on $[0, 2 π]$ .

Find $f^{''} (x)$ :

$f^{'} (x) = cos (x)$

$f^{''} (x) = - sin (x)$

Find where $f^{''} (x) = 0$ :

$- sin (x) = 0$

$x = 0, π, 2 π$

Test concavity:

For $0 < x < π$ : $f^{''} (\frac{π}{2}) = - 1 < 0$ (concave down)
For $π < x < 2 π$ : $f^{''} (\frac{3 π}{2}) = 1 > 0$ (concave up)

Concavity:

Concave down on $(0, π)$
Concave up on $(π, 2 π)$

Inflection point: $x = π$

Taken from Desmos

Relationship Between f, f', and f''

Complete Analysis

Understanding the relationships:

$f^{'} > 0$ → $f$ increasing
$f^{'} < 0$ → $f$ decreasing
$f^{''} > 0$ → $f$ concave up (and $f^{'}$ increasing)
$f^{''} < 0$ → $f$ concave down (and $f^{'}$ decreasing)

Example 7:

For $f (x) = x^{3} - 3 x$ , describe the behavior at $x = 1$ .

Find derivatives:

$f^{'} (x) = 3 x^{2} - 3$

$f^{''} (x) = 6 x$

At $x = 1$ :

$f (1) = 1 - 3 = - 2$

$f^{'} (1) = 3 - 3 = 0$

$f^{''} (1) = 6 > 0$

Interpretation:

$f^{'} (1) = 0$ means horizontal tangent (critical point)
$f^{''} (1) > 0$ means concave up
Since the function is concave up with a horizontal tangent, this is a local minimum

FiveHive

FiveHive

5.6 - Determining Concavity of Functions over Their Domains

Introduction

Understanding Concavity

Visual and Mathematical Definitions

Using the Second Derivative to Determine Concavity

The Second Derivative Test for Concavity

Finding Intervals of Concavity

Points of Inflection

Definition and Identification

When Points Are Not Inflection Points

Analyzing More Complex Functions

Functions with Multiple Inflection Points

Trigonometric Functions

Relationship Between f, f', and f''

Complete Analysis

Practice Section