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Concavity of
Functions

Learn how the second derivative reveals whether a curve bends upward or downward, and master the art of determining concavity over entire domains.

AP Calculus AB

What Is Concavity?

Concavity describes the direction a curve bends. Imagine driving along a road: concavity tells you whether the road is curving like a bowl (holding water) or an umbrella (shedding water).

Concave Up
The curve opens upward like a cup. The slope is increasing.
Concave Down
The curve opens downward like a frown. The slope is decreasing.

Visualizing Concavity

Concave Up Tangent slopes increase Concave Down Tangent slopes decrease

Why Does Concavity Matter?

Concavity is essential in calculus because it tells us about the behavior and shape of functions beyond what the first derivative reveals.

Second Derivative Test
Classify critical points as local max or local min without a sign chart.
Curve Sketching
Know the exact shape of a graph between key points.
Optimization
Verify whether an extreme value is a maximum or minimum.

The Second Derivative Test

The sign of \( f''(x) \) directly determines the concavity. Think of it this way: \( f'' \) measures the rate of change of the slope. Positive means the slope is speeding up (bending up); negative means the slope is slowing down (bending down).

Inflection Points: Where Concavity Changes

x y Inflection Point Concave Down Concave Up

Increasing/Decreasing vs. Concavity

Increasing/Decreasing Concavity
Derivative used \( f'(x) \) \( f''(x) \)
What it measures Direction of \( f \) Bending of \( f \)
Positive sign \( f \) is rising \( f \) bends up
Negative sign \( f \) is falling \( f \) bends down
Changes at Critical points Inflection points

A continuous route through hills: elevation, grade, and intermediate height

A walking path crosses a ridge and hollow beside a staff-bearing observer.For f(x) = x³ − 3x, height, slope, and change in slope are different quantities. The extrema occur at x = −1 and 1; concavity changes at x = 0.

Height, grade and changing grade

For f(x) = x³ − 3x, height, slope, and change in slope are different quantities. The extrema occur at x = −1 and 1; concavity changes at x = 0.For f(x) = x³ − 3x, height, slope, and change in slope are different quantities. The extrema occur at x = −1 and 1; concavity changes at x = 0.

The Four Shape Combinations

Increasing + Concave Up
\( f' > 0 \)
\( f'' > 0 \)
 ↗ Increasing + Concave Down
\( f' > 0 \)
\( f'' < 0 \)
 ↗ Decreasing + Concave Up
\( f' < 0 \)
\( f'' > 0 \)
 ↘ Decreasing + Concave Down
\( f' < 0 \)
\( f'' < 0 \)
 ↙

How to Determine Concavity

1
Find \( f''(x) \). Take the second derivative of \( f(x) \).
2
Solve \( f''(x) = 0 \). Also find where \( f''(x) \) is undefined. These are candidate inflection points.
3
Build a sign chart. Test the sign of \( f''(x) \) in each interval between candidates.
4
State concavity. Positive intervals are concave up; negative intervals are concave down.
5
Confirm inflection points. A sign change in \( f'' \) means an inflection point exists there.

Example 1: \( f(x) = x^3 - 3x^2 + 2 \)

1
First derivative: \( f'(x) = 3x^2 - 6x \)
2
Second derivative: \( f''(x) = 6x - 6 \)
3
Set \( f''(x) = 0 \): \( 6x - 6 = 0 \implies x = 1 \)

Example 1: Graph

-1 0 1 2 3 2 0 -2 Inflection (1, 0) Concave Down Concave Up

Example 2: \( g(x) = x^4 - 4x^3 \)

1
First derivative: \( g'(x) = 4x^3 - 12x^2 \)
2
Second derivative: \( g''(x) = 12x^2 - 24x = 12x(x - 2) \)
3
Zeros of \( g'' \): \( x = 0 \) and \( x = 2 \)

Example 3: \( h(x) = \sin(x) \) on \( [0, 2\pi] \)

1
Second derivative: \( h''(x) = -\sin(x) \)
2
Set \( h''(x) = 0 \): \( -\sin(x) = 0 \implies x = 0, \pi, 2\pi \)
0
\( \frac{\pi}{2} \)
\( \pi \)
\( \frac{3\pi}{2} \)
\( 2\pi \)
 Concave Down Concave Up
Inflection at \( x = \pi \)

Special Case: \( f''(x) = 0 \) Without an Inflection Point

Consider \( f(x) = x^4 \). We have \( f''(x) = 12x^2 \), so \( f''(0) = 0 \). But \( f''(x) \geq 0 \) for all \( x \), meaning the concavity never changes. There is no inflection point at \( x = 0 \).

x y x = 0 Always Concave Up

Common Mistakes

✕ Assuming \( f''(c) = 0 \) means inflection
You must confirm a sign change in \( f'' \). Example: \( f(x) = x^4 \) has \( f''(0) = 0 \) but no inflection point.
✕ Confusing concavity with direction
Concave up does not mean increasing. A function can be decreasing and concave up simultaneously.
✕ Forgetting the domain
Always check where \( f'' \) is undefined. Points where \( f'' \) doesn't exist can also be inflection candidates.

Concavity Summary

Condition Concavity Graph Shape
\( f''(x) > 0 \) Concave Up Opens upward (cup)
\( f''(x) < 0 \) Concave Down Opens downward (frown)
\( f''(x) = 0 \) + sign change Inflection Point Concavity switches
\( f''(x) = 0 \), no sign change Same concavity No inflection

Question 1 of 5

On what interval is \( f(x) = x^3 - 6x^2 + 9x + 1 \) concave up?

Question 2 of 5

If \( f''(x) = x^2(x - 3) \), at which \( x \)-value does \( f \) have an inflection point?

Question 3 of 5

The function \( g(x) = e^{-x^2} \) is concave down on which interval?

Question 4 of 5

If a function \( f \) is increasing and concave down on an interval, what must be true about \( f'(x) \) and \( f''(x) \)?

Question 5 of 5

How many inflection points does \( f(x) = x^4 - 6x^2 + 8x + 3 \) have?