Reading Concavity from a First-Derivative Graph

A graph of \(f'\) contains two kinds of visual information. Its height tells you whether \(f\) is increasing or decreasing; its direction of change tells you whether \(f\) is concave up or concave down. Keeping those ideas separate is essential when reading derivative graphs.

For concavity, focus on what happens to the values of \(f'\) as \(x\) moves from left to right. If those values rise, the slopes of \(f\) are getting larger, so \(f\) is concave up. If they fall, the slopes of \(f\) are getting smaller, so \(f\) is concave down. A change from rising to falling, or from falling to rising, marks a possible inflection point.

Key idea: On a graph of \(y=f'(x)\), the direction the graph moves as \(x\) increases indicates the concavity of \(f\). The graph’s height, by itself, does not.

What Concavity Means Here

A function is concave up on an interval when its slopes increase as \(x\) increases. Geometrically, the graph bends upward: its tangent slopes become more positive, or less negative. A function is concave down when its slopes decrease, so its graph bends downward.

In terms of \(f'\), this means that a nondecreasing first derivative corresponds to concavity up, while a nonincreasing first derivative corresponds to concavity down. The word “nondecreasing” allows \(f'\) to be constant on part of an interval. A constant \(f'\) means that \(f\) is linear there, so the graph has no bending on that portion.

Definition: On an interval where \(f\) is differentiable, \(f\) is concave up if \(f'\) is nondecreasing, and concave down if \(f'\) is nonincreasing. If \(f'\) is strictly increasing or strictly decreasing, the corresponding concavity is strict.

When using a graph, read it from left to right. An upward-sloping portion of the \(f'\) graph means \(f'\) is increasing, so \(f\) is concave up. A downward-sloping portion means \(f'\) is decreasing, so \(f\) is concave down. A horizontal portion means \(f'\) is constant and \(f\) is linear there.

Why the Direction of \(f'\) Determines Concavity

The connection follows from how secant slopes of \(f\) compare. Take three numbers \(a<b<c\) in an interval where \(f\) is differentiable and \(f'\) is nondecreasing. The Mean Value Theorem applied on \([a,b]\) gives a number \(u\) between \(a\) and \(b\) such that the slope of the secant line on that interval is \(f'(u)\). Applied on \([b,c]\), it gives a number \(v\) between \(b\) and \(c\) such that the second secant slope is \(f'(v)\).

\frac{f(b)-f(a)}{b-a}=f'(u), \qquad \frac{f(c)-f(b)}{c-b}=f'(v).

Since \(u<b<v\), we have \(u<v\). Because \(f'\) is nondecreasing, \(f'(u)\leq f'(v)\). Thus secant slopes increase as we move from the left interval to the right interval, which is the defining geometric behavior of a concave-up function. If \(f'\) is nonincreasing, the same reasoning reverses the inequality and gives concavity down.

Theorem: If \(f\) is differentiable on an interval and \(f'\) is nondecreasing there, then \(f\) is concave up on that interval. If \(f'\) is nonincreasing, then \(f\) is concave down. This result requires the stated behavior of \(f'\) across the interval, not merely at one point.

This reasoning explains why the graph of \(f'\) is useful: it displays the way the slopes of \(f\) change. You do not need to know the exact formula for \(f\) to read that qualitative information.

A Reliable Reading Process

Read left to right.
Identify the intervals on which the graph of \(f'\) rises, falls, or stays horizontal.

Translate each direction.
Rising \(f'\) means \(f\) is concave up; falling \(f'\) means \(f\) is concave down. Constant \(f'\) means \(f\) is linear on that interval.

Check each direction change.
A switch from rising to falling, or falling to rising, indicates a possible change in concavity.

Check the original function at the candidate.
An inflection point must be a point on the graph of \(f\), so the function must be defined there. Report the \(x\)-coordinate if the graph of \(f'\) does not provide the corresponding value of \(f\).

Worked Examples

Worked Example: A Derivative Graph with a Minimum

Suppose the graph of \(f'\) on \([-2,2]\) is \(f'(x)=x^2\), with \(f(0)=0\). Determine the concavity of \(f\) and any inflection point.

First read the derivative graph’s direction. The values of \(x^2\) decrease as \(x\) moves from \(-2\) to \(0\), then increase as \(x\) moves from \(0\) to \(2\). For example, the graph passes through \((-2,4)\), \((0,0)\), and \((2,4)\).

\begin{aligned} -2<x<0 &: f' \text{ is decreasing, so } f \text{ is concave down},\\ 0<x<2 &: f' \text{ is increasing, so } f \text{ is concave up}. \end{aligned}

The direction changes at \(x=0\), so \(x=0\) is a possible inflection point. To find the point’s coordinates, recover \(f\) from its derivative:

f(x)=\int x^2\,dx=\frac{x^3}{3}+C.

Using \(f(0)=0\) gives \(C=0\), so \(f(x)=x^3/3\). The candidate point is \((0,0)\). Check by differentiating: \(f'(x)=x^2\), as given. The derivative graph changes from decreasing to increasing at \(0\), and \(f\) is defined there, so \((0,0)\) is an inflection point.

Worked Example: Rising, Flat, Then Falling

Suppose \(f(0)=1\), and the graph of \(f'\) consists of line segments given by \(f'(x)=x\) on \([0,2]\), \(f'(x)=2\) on \([2,4]\), and \(f'(x)=6-x\) on \([4,6]\). Find the concavity intervals and any inflection point.

On \((0,2)\), \(f'\) rises from \(0\) to \(2\). On \((2,4)\), it stays at \(2\). Together, \(f'\) is nondecreasing from \(0\) to \(4\), so \(f\) is concave up on \((0,4)\), with a linear portion on \((2,4)\). On \((4,6)\), \(f'\) falls from \(2\) to \(0\), so \(f\) is concave down there. The change from nondecreasing to decreasing at \(x=4\) gives a possible inflection point.

To find its coordinates, integrate each piece in turn while matching the function values at the joins. On \([0,2]\),

f(x)=1+\int_0^x t\,dt=1+\frac{x^2}{2}, \qquad f(2)=1+\frac{2^2}{2}=3.

On \([2,4]\), use \(f(2)=3\):

f(x)=3+\int_2^x 2\,dt =3+2(x-2)=2x-1, \qquad f(4)=7.

On \([4,6]\), use \(f(4)=7\):

\begin{aligned} f(x)&=7+\int_4^x(6-t)\,dt\\ &=7+\left(6x-\frac{x^2}{2}\right)-\left(24-8\right)\\ &=6x-\frac{x^2}{2}-9. \end{aligned}

Check the reconstruction by differentiating: the three formulas give \(x\), \(2\), and \(6-x\), respectively, which match the specified \(f'\). The values also match at the joins: \(f(2)=3\) from both adjacent formulas, and \(f(4)=7\) from both. Since \(f(4)=7\) and concavity changes there, the inflection point is \((4,7)\). At \(x=2\), \(f'\) changes from rising to flat but does not switch to falling; concavity does not change from up to down or down to up there.

Worked Example: A Horizontal Tangent on the Derivative Graph

Suppose \(f'(x)=x^3\) on \((-2,2)\), and \(f(0)=0\). The derivative graph has a horizontal tangent at \(x=0\). Does \(f\) have an inflection point there?

For negative \(x\), \(x^3\) increases as \(x\) increases toward \(0\). For positive \(x\), \(x^3\) continues to increase. Thus \(f'\) is increasing on the entire interval, so \(f\) is concave up on \((-2,2)\). Although the derivative graph flattens at \(0\), it does not change from increasing to decreasing or from decreasing to increasing.

We can verify by finding \(f\):

f(x)=\int x^3\,dx=\frac{x^4}{4}+C.

The condition \(f(0)=0\) gives \(C=0\). Differentiating \(f(x)=x^4/4\) gives \(f'(x)=x^3\), as required. The function is concave up on both sides of \(0\), so there is no inflection point at \(x=0\). A horizontal tangent on the graph of \(f'\) alone is not enough to establish a change in concavity.

Finding Possible Inflection Points

An inflection point is a point on the graph of \(f\) where concavity changes. On a graph of \(f'\), look for a change in its direction: rising to falling indicates a change from concave up to concave down; falling to rising indicates a change from concave down to concave up.

These locations are candidates, not automatic conclusions. Confirm that \(f\) is defined at the corresponding \(x\)-value. Also check the behavior on both sides: merely reaching a high or low point on the derivative graph is not sufficient if its direction does not actually change. When only the graph of \(f'\) is provided, it may identify the inflection point’s \(x\)-coordinate without giving the value \(f(x)\).

Important distinction: A zero of \(f'\) concerns the height of the derivative graph. A possible inflection point concerns a change in its direction. Neither the \(x\)-axis crossing nor a horizontal tangent on the \(f'\) graph, by itself, establishes an inflection point.

A Common Mistake: Using the Height Instead of the Direction

A common error is to conclude that \(f\) is concave up wherever \(f'\) is positive. But a positive value of \(f'\) says that \(f\) is increasing; it does not say that \(f'\) is increasing.

For a concrete counterexample, let \(f'(x)=-x\) on \((-1,1)\). Since \(-x>0\) for \(-1<x<0\), the mistaken reasoning would say that \(f\) is concave up on that interval. However, the graph of \(f'\) is decreasing throughout \((-1,1)\), so \(f\) is concave down throughout. Indeed, integrating gives \(f(x)=-x^2/2+C\), and differentiating twice gives \(f''(x)=-1\), confirming concavity down.

Correction: Use the sign, or height, of \(f'\) to determine whether \(f\) increases or decreases. Use whether \(f'\) rises or falls to determine whether \(f\) is concave up or down.

Why This Reading Skill Matters

Derivative graphs often reveal the shape of a function even when the function itself is difficult to calculate or is not given explicitly. Concavity tells you how the function’s slopes evolve, and changes in concavity help locate possible inflection points. These observations can support a curve analysis without confusing the behavior of \(f\) with the behavior of \(f'\).

When you work from a derivative graph, state the intervals and the direction you observed. For example, “\(f'\) is decreasing on \((a,b)\), so \(f\) is concave down there” makes the evidence and conclusion clear. At candidate inflection points, explicitly compare the behavior on the left and right.

Check Your Understanding

Use the direction of the graph of \(f'\) to answer each question.

If \(f'\) is increasing on \((1,5)\), what is the concavity of \(f\) on that interval?
If \(f'\) decreases on \((-3,0)\) and increases on \((0,3)\), what does this suggest about \(x=0\)? What must still be checked?
If \(f'\) is positive but decreasing on an interval, what can you conclude about whether \(f\) is increasing and about its concavity?
Why does a horizontal tangent on the graph of \(f'\) not necessarily identify an inflection point?
If \(f'\) rises on \((a,b)\), stays constant on \((b,c)\), and falls on \((c,d)\), where does the concavity of \(f\) change, if anywhere?

Reading Concavity from a First-Derivative Graph

What You'll Learn