hilly shapes representing concave down definition

Calculus

Concave Up and Concave Down: Meaning and Examples

04.12.2022 • 8 min read

Rachel McLean

Subject Matter Expert

In this article, we’ll learn the definition of concavity. Using graphs, we’ll compare concave up vs. concave down curves. Then, we’ll discuss how to find points of inflection and identify intervals of concave up and concave down, using examples.

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In This Article

  1. What Is Concavity?

  2. What Is a Point of Inflection?

  3. How to Find Intervals of Concave Up and Concave Down

What Is Concavity?

First derivatives tell us very useful information about the behavior of a function. First derivatives are used to determine if a function is increasing, decreasing or constant on an interval. You are probably already familiar with how to test if f(x)f(x) is increasing, decreasing, or constant. Suppose a function ff is differentiable on the interval II. Then:

  • If f(x)>0f’(x) > 0 for each xx on II, then ff is increasing on II.

  • If f’(x) < 0 for each x on I, then f is decreasing on I.

  • If f(x)=0f’(x) = 0 for each xx on II, then ff is constant on II.

concave graph testing if f(x) is increasing, decreasing, or constant

We can apply a very similar way of thinking to second derivatives. Like first derivatives, second derivatives also tell us useful information about the behavior of a function. You can find a second derivative—usually denoted by f’’f’’—by taking the derivative of the first derivative.

The first derivative ff’ can tell us when ff is increasing or decreasing, while the second derivative f’’f’’ can tell us the shape of the graph ff. More specifically, f’’f’’ tells us the concavity of a graph: whether the graph of ff is concave up or concave downward.

Concave up intervals look like valleys on a graph, while concave down intervals look like mountains. It might be helpful to visualize that concave up intervals could hold water, while concave down intervals could not hold water. Concave downward curves are also referred to as “concave curves” and concave up curves are also referred to as “convex curves.”

Study the graphs below to visualize examples of concave up vs concave down intervals.

2 graph examples of concave up vs concave down intervals
1 graph with examples of concave up, concave down, and no concavity intervals

It’s important to keep in mind that concavity is separate from the notion of increasing/decreasing/constant intervals. A concave up interval can contain both increasing and/or decreasing intervals. A concave downward interval can contain both increasing and/or decreasing intervals.

Graph showing that a concave up interval and a concave down interval can contain both increasing and/or decreasing intervals

Remember that the first derivative ff’ gives us the rate of change of the function ff, which allows us to determine when ff is increasing, decreasing, or constant. Similarly, the second derivative f’’f’’ gives us the rate of change of the first derivative ff’, which allows us to determine when ff’ is increasing, decreasing, or constant.

So, if ff is a differentiable function on the interval II with derivatives ff’ and f’’f’’, the concave up and concave down definition is:

  • If f’’(x)>0f’’(x) > 0 for each xx on II, then ff’ is increasing on II, and ff is concave up on II.

  • If f’’(x) < 0 for each x on I, then f’ is decreasing on I, and f is concave down on I.

  • If f’’(x)=0f’’(x) = 0 for each xx on II, then ff’ is constant on II, and ff has no concavity.

We can separate the above statements into two ways to test for the concavity of a function on an interval II:

  • If ff’ is increasing on II, then ff is concave up on II.

  • If ff’ is decreasing on II, then ff is concave down on II.

  • If ff’ is constant on II, then ff has no concavity.

Or:

  • If f’’(x)>0f’’(x) > 0 for each xx on II, then ff is concave up on II.

  • If f’’(x) < 0 for each x on I, then f is concave down on I.

  • If f’’(x)=0f’’(x) = 0 for each xx on II, then ff has no concavity.

So, when the second derivative is positive, the graph is concave up. When the second derivative is negative, the graph is concave down. When the second derivative is zero, the graph has no concavity.

If a graph has no concavity on II, that means that the graph of ff is linear over the interval II.

Graph showing that if a graph has no concavity on I, then the graph of f is linear over the interval I.

Dr. Hanna Fry dives more into a function’s concavity in this lesson clip, as well as a first derivative test, some terminology, and a second derivative test.

What Is a Point of Inflection?

Inflection points are points where ff changes concavity. These points occur where f’’(x)f’’(x) changes sign. If xx is an inflection point of ff, then f’’(x)=0f’’(x) = 0 or f’’(x)f’’(x) is undefined.

To find all inflection points, you need to consider all points where f’’(x)=0f’’(x) = 0 or where f’’(x)f’’(x) is undefined. These are possible inflection points, but not guaranteed inflection points.

graph showing an inflection point

To verify that a point xx is an inflection point, you must check that the values of f’’(x)f’’(x) to the left and right of xx have opposite signs.

For example, suppose we are given the function f(x)=x33x2+1f(x) = x^3 - 3 x^2 + 1, and suppose we are asked to check if x=1x = 1 is an inflection point. First, we must calculate the first and second derivatives. Using the power rule, we find that f(x)=3x26xf’(x) = 3x^2-6x and f’’(x)=6x6f’’(x) = 6x-6.

Plugging x=1x = 1 into f’’(x)=6x6f’’(x) = 6x-6, we get f’’(1)=66=0f’’(1) = 6 - 6 = 0. So, x=1x = 1 is a possible inflection point. Now, we need to check that the values of f’’(x)f’’(x) to the left and right of x=1x = 1 have opposite signs.

Let’s choose one point to the left of x=1x = 1, say x=0x = 0. Plugging x=0x = 0 into f’’(x)=6x6f’’(x) = 6x-6, we find that f’’(0)=06=6f’’(0) = 0 - 6 = -6. So, the values to the left of x=1x = 1 are negative.

Let’s choose one point to the right of x=1x = 1, say x=2x = 2. Plugging x=2x = 2 into f’’(x)=6x6f’’(x) = 6x-6, we find that f’’(2)=126=6f’’(2) = 12 - 6 = 6. So, the values to the right of x=1x = 1 are positive.

It’s helpful to visualize this process with a number line.

Number line showing concave down, concave up, and 1 = inflection point

Since the values of f’’(x)f’’(x) to the left and right of x=1x = 1 have opposite signs, x=1x = 1 is an inflection point.

A point that is not an inflection point would have the same sign on both sides of the dotted line. For example, x=1x = -1 is clearly not an inflection point, since values to the left and right of f’’(1)f’’(-1) are both negative and f’’(1)0f’’(-1) \not = 0.

Number line showing concave down, concave up, and -1 = inflection point

We can verify that x=1x = 1 is an inflection point by looking at the graph of ff.

A graph of f where we can verify that x = 1 is an inflection point

So, how can we find all the inflection points of a function? Finding all inflection points of a function reveals the intervals where a function is concave up and concave down, so you’ll learn these steps in the next section.

How to Find Intervals of Concave Up and Concave Down

Now that we have the rules below, how can we find all intervals where a function ff is concave up or concave down?

  • If f’’(x)>0f’’(x) > 0 for each xx on II, then ff is concave up on II.

  • If f’’(x) < 0 for each x on I, then f is concave down on I.

  • If f’’(x)=0f’’(x) = 0 for each xx on II, then ff has no concavity.

6 Steps to Determine Concavity

Given a function ff, here are 6 simple steps to determine its concavity:

Step 1

Differentiate to find f(x)f’(x).

Step 2

Differentiate f(x)f’(x) to find f’’(x)f’’(x).

Step 3

Factor f’’(x)f’’(x) to find the x-values where f’’(x)=0f’’(x) = 0. Also determine the x-values where f’’(x)f’’(x) is undefined, if any.

Step 4

Draw a number line, marking the x-values found in the step above.

Step 5

For each number marked on the line, choose one “test” x-value to the left and one “test” x-value to the right. Plug these test x-values into f’’(x)f’’(x) and record the resulting sign (negative or positive) in the table.

Step 6

Any marked x-value that has opposite signs to the left and right is an inflection point. Any marked x-value that has the same sign to the left and right is not an inflection point.

Find Concavity & Inflection Points Example

Let’s do one example together. Find all intervals where f(x)=x42x3f(x) = \frac{x^4}{2}-x^3 is concave up or down, and find all inflection points.

Step 1

Using the power rule, f(x)f’(x)= 2x33x22x^3 - 3x^2.

Step 2

Using the power rule, f’’(x)=6x26xf’’(x) = 6x^2 - 6x.

Step 3

Factoring f’’(x)f’’(x), we find f’’(x)=6(x1)xf’’(x) = 6(x-1)x. So f’’(x)=0f’’(x) = 0 when x=1x = 1 and x=0x = 0.

Step 4

Now, we can create our table.

building a concavity table

Step 5

Choosing a point to the left of x=0x = 0, say x=1x = -1, we find that f’’(1)=6+6=12f’’(-1) = 6 + 6 = 12. So, we can put a plus sign to the left of x=0x = 0. Since f’’(x)>0f’’(x) > 0, this means that ff is concave up on the interval (,0)(- \infty, 0).

Choosing a point to the right of x = 0, say x = \frac{1}{2}, we find that f’’(\frac{1}{2}) = \frac{3}{2} - 3 = -\frac{1}{2}. Since f’’(x) < 0, this means that f is concave down on the interval (0, 1).

Choosing a point to the right of x=1x = 1, say x=2x = 2, we find that f’’(2)=2412=12f’’(2) = 24 - 12 = 12. Since f’’(x)>0f’’(x) > 0, this means that ff is concave up on the interval (1,)(1, \infty).

Table showing that f is concave up on the interval (- \infty, 0) and (1, \infty), and concave down on the interval (0, 1).

Step 6

Since both x=0x = 0 and x=1x = 1 have opposite signs to their left and right, both x=0x = 0 and x=1x = 1 are inflection points.

Analyzing our table, we can see that ff is concave up on the interval (,0)(- \infty, 0) and (1,)(1, \infty), and concave down on the interval (0,1)(0, 1).

We can verify our answer by looking at the graph of f(x)=x42x3f(x) = \frac{x^4}{2}-x^3:

Graph where f is concave up and concave down

Now, let’s learn how to find concavity from the first derivative test graph. Remember we deduced earlier that we can also determine concavity using these rules:

  • If ff’ is increasing on II, then ff is concave up on II.

  • If ff’ is decreasing on II, then ff is concave down on II.

  • If ff’ is constant on II, then ff has no concavity.

As long as you have the graph of ff’, you can visually determine where ff is concave up or down by identifying the intervals where ff’ is increasing.

For example, consider the function f(x)=x48x3+18x2f(x) = x^4 - 8x^3 + 18x^2. The graph of its first derivative f(x)=4x324x2+36xf’(x) = 4x^3 - 24x^2 + 36x is given below.

Graph showing f’ increasing on the intervals (-\infty, 1) and (3, \infty), and decreasing on the interval (1, 3)

Using the graph, ff’ is increasing on the intervals (,1)(-\infty, 1) and (3,)(3, \infty), and decreasing on the interval (1,3)(1, 3).

Since ff is concave up when ff’ is increasing and concave down when ff’ is decreasing, then ff is concave up on the intervals (,1)(-\infty, 1) and (3,)(3, \infty), and concave down on the interval (1,3)(1, 3).

We can verify our answer by looking at the graph of ff.

Graph  showing f is concave up when f’ is increasing and concave down when f’ is decreasing. Then f is concave up on the intervals (-\infty, 1) and (3, \infty), and concave down on the interval (1, 3)

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