5.6 - Determining Concavity of Functions over Their Domains

Understanding Concavity

Concavity describes which way a curve bends.

Concave up: opens upward like a cup that holds water,

\cup

Concave down: opens downward like an upside-down cup,

\cap

No concavity: a straight line has none

5.6 - Determining Concavity of Functions over Their Domains

Concavity and the First Derivative

When a function is concave up, the slope (the first derivative) is increasing.

Moving left to right, the slope goes negative

\rightarrow

zero

\rightarrow

positive. The derivative values are increasing.

When a function is concave down, the slope is decreasing.

The Second Derivative Test for Concavity

f'(x)

increasing means

(f'(x))' > 0

, and

(f'(x))' = f''(x)

, then:

\text{If } f''(x) > 0, \text{ then } f \text{ is concave up}

\text{If } f''(x) < 0, \text{ then } f \text{ is concave down}

Mnemonic: positive second derivative is a happy face

\cup

(concave up). Negative second derivative is a sad face

\cap

(concave down).

Finding Intervals of Concavity

Find $f''(x)$

Solve $f''(x) = 0$ and find where $f''(x)$ doesn’t exist

Make a sign chart

Test the sign of $f''(x)$ in each interval

Interpret: positive means concave up, negative means concave down

Example 1

Find the intervals of concavity for

g(x) = x^4 - 8x^3 + 18x^2 - 5

Step 1: First derivative.

g'(x) = 4x^3 - 24x^2 + 36x

Step 2: Second derivative.

g''(x) = 12x^2 - 48x + 36

Step 3: Solve

g''(x) = 0

12(x^2 - 4x + 3) = 0 \quad \Rightarrow \quad 12(x - 1)(x - 3) = 0 \quad \Rightarrow \quad x = 1 \text{ or } x = 3

Step 4: Sign chart.

Interval	$(-\infty, 1)$	$x = 1$	$(1, 3)$	$x = 3$	$(3, \infty)$
Sign of $g''(x)$	$(+)$	$0$	$(-)$	$0$	$(+)$

Test points:

$x = 0$ : $g''(0) = 36 > 0$

$x = 2$ : $g''(2) = 48 - 96 + 36 = -12 < 0$

$x = 4$ : $g''(4) = 192 - 192 + 36 = 36 > 0$

Step 5: Conclusions.

Concave up on

(-\infty, 1) \cup (3, \infty)

because

g''(x) > 0

Concave down on

(1, 3)

because

g''(x) < 0

Points of Inflection

A point of inflection happens at

x = c

where:

$f$ is continuous at $c$

The concavity changes (which means $f''(x)$ changes sign)

The key idea: a point of inflection is where the graph changes concavity.

Back to Example 1

Points of inflection at

x = 1

and

x = 3

because

g''(x)

changes sign there.

At $x = 1$ : changes from $(+)$ to $(-)$

At $x = 3$ : changes from $(-)$ to $(+)$

Justification: points of inflection exist because

g''(x)

changes sign.

Common Mistakes

Wrong: “If

f''(c) = 0

, then

x = c

is a point of inflection.”

Right: “

x = c

is a candidate for a point of inflection. You have to verify the sign change.”

Wrong: “If

f''(c)

doesn’t exist, there can’t be a point of inflection.”

Right: “Even if

f''(c)

doesn’t exist, there can still be a point of inflection as long as concavity changes.”

Point of Inflection Where $f''$ Doesn’t Exist

All you need for a point of inflection:

$f$ must be continuous at $c$

Concavity must change

The second derivative doesn’t have to exist.

Reading from the Graph of $f'(x)$

Given the graph of

f'(x)

Concave up where

f'(x)

is increasing (the slope of

f'

is positive)

Concave down where

f'(x)

is decreasing (the slope of

f'

is negative)

Example 2

Where is

f'(x)

decreasing? On

(-4, -1)

and

(1, 2)

Where is

f'(x)

increasing? On

(-1, 1)

and

(2, 3)

Conclusions about $f(x)$ :

Concave down on

(-4, -1) \cup (1, 2)

because

f'(x)

is decreasing.

Concave up on

(-1, 1) \cup (2, 3)

because

f'(x)

is increasing.

Points of inflection at

x = -1, 1, 2

because concavity changes there.

Reading from the Graph of $f''(x)$

Given the graph of the second derivative:

Concave up where

f''(x) > 0

(above the x-axis)

Concave down where

f''(x) < 0

(below the x-axis)

Points of inflection where

f''(x)

crosses the x-axis (changes sign)

Example 3

Where is

f''(x) > 0

? Above the axis:

(-4, -3) \cup (1, 3)

Where is

f''(x) < 0

? Below the axis:

(-3, 1) \cup (3, 5)

Concave up on

(-4, -3) \cup (1, 3)

Concave down on

(-3, 1) \cup (3, 5)

Points of inflection at

x = -3, 1, 3

where

f''(x)

crosses the x-axis

Application: Tangent Line Estimates

Concavity tells you whether a tangent line approximation is an overestimate or an underestimate.

If $f''(x) > 0$ (concave up), the tangent line lies below the curve, so it’s an underestimate

If $f''(x) < 0$ (concave down), the tangent line lies above the curve, so it’s an overestimate

Example 4

Does the tangent line to

h(x) = xe^{-x}

x = 1

lie above or below the curve?

First derivative (product rule):

h'(x) = e^{-x} + x e^{-x}(-1) = e^{-x}(1 - x)

Second derivative:

h''(x) = -e^{-x}(1 - x) + e^{-x}(-1) = -e^{-x} + xe^{-x} - e^{-x} = e^{-x}(x - 2)

Evaluate at $x = 1$ :

h''(1) = e^{-1}(1 - 2) = -\frac{1}{e} < 0

Since

h''(1) < 0

, the function is concave down at

x = 1

, so the tangent line there lies above the curve. It’s an overestimate.

Side note: at

x = 2

h''(2) = e^{-2}(0) = 0

, so that’s a candidate for a point of inflection, not a place where you can call the tangent line an over or underestimate.

Key Takeaways

Concave up: $f''(x) > 0$ (happy face $\cup$ )

Concave down: $f''(x) < 0$ (sad face $\cap$ )

Concave up is the same as saying $f'(x)$ is increasing

Concave down is the same as saying $f'(x)$ is decreasing

A point of inflection happens where concavity changes, meaning $f''(x)$ changes sign

$f''(c) = 0$ does NOT guarantee a point of inflection, you have to verify the sign change

A point of inflection can exist even if $f''(c)$ is undefined

From the graph of $f'(x)$ : look at where $f'$ increases or decreases

From the graph of $f''(x)$ : look at where it’s above or below the x-axis

Tangent line estimates: concave up means underestimate, concave down means overestimate

Back Next

5.6 - Determining Concavity of Functions over Their Domains

Understanding Concavity

Concavity and the First Derivative

The Second Derivative Test for Concavity

Finding Intervals of Concavity

Example 1

Points of Inflection

Back to Example 1

Common Mistakes

Point of Inflection Where f′′f''f′′ Doesn’t Exist

Reading from the Graph of f′(x)f'(x)f′(x)

Example 2

Reading from the Graph of f′′(x)f''(x)f′′(x)

Example 3

Application: Tangent Line Estimates

Example 4

Key Takeaways

Point of Inflection Where $f''$ Doesn’t Exist

Reading from the Graph of $f'(x)$

Reading from the Graph of $f''(x)$