Introduction
A point where the second derivative is zero often attracts attention during curve analysis. A point where the second derivative is undefined does too. Both can be important places to investigate, but neither fact alone establishes that the graph has an inflection point there.
The central question is whether the graph’s concavity changes across the point. This is a different question from whether the function has a local maximum or minimum. The two kinds of behavior can occur together, but neither implies the other.
What Counts as an Inflection Point?
Concavity describes how the graph bends. On an interval where \(f\) is twice differentiable, a positive second derivative indicates that the graph is concave up, while a negative second derivative indicates that it is concave down. Equivalently, where the derivatives exist, \(f'\) is increasing on an interval of concavity up and decreasing on an interval of concavity down.
For this tutorial, an inflection point occurs at an interior point \(x=c\) of the domain if \(f\) is continuous at \(c\) and the graph changes concavity across \(c\): it is concave up on one side and concave down on the other, or vice versa, in sufficiently small neighboring intervals. The point on the graph is \((c,f(c))\).
In many examples, the sign of \(f''\) gives a practical test. If \(f''(x)<0\) just to the left of \(c\) and \(f''(x)>0\) just to the right, concavity changes from down to up. Reversing those signs gives a change from up to down. If the second derivative has the same sign on both sides, there is no such change, even if \(f''(c)=0\) or \(f''(c)\) is undefined.
Why a Zero Second Derivative Is Not Enough
A common error is to solve \(f''(x)=0\), find a value \(c\), and immediately declare \(c\) an inflection point. The equation \(f''(c)=0\) gives a candidate, not a conclusion. The sign of \(f''\) must still be checked on both sides.
Worked Example: A Zero Second Derivative Without an Inflection
Consider \(f(x)=x^4\). Determine whether \(x=0\) is an inflection point.
First calculate the derivatives:
At \(x=0\), the second derivative is
Now check the sign on each side. For every \(x<0\), \(x^2>0\), so \(12x^2>0\). For every \(x>0\), \(x^2>0\), so \(12x^2>0\) there as well. Thus \(f''\) is positive on both neighboring intervals: the graph is concave up on both sides of zero.
Therefore, concavity does not change at zero, so \((0,0)\) is not an inflection point. As a check of the point on the graph, \(f(0)=0^4=0\). The zero second derivative was a candidate, but the sign test rules out an inflection.
The key distinction is between a second derivative’s value at one point and its behavior on intervals around that point. Concavity is an interval property; a single value cannot reveal whether the bending switches direction.
A Sign Change Can Give an Inflection Without an Extremum
Concavity and local extrema describe different features. A local maximum or minimum concerns whether nearby function values lie below or above the value at a point. An inflection point concerns a change in concavity. A graph can change concavity while continuing to increase, or while continuing to decrease.
Worked Example: An Inflection Point That Is Not a Local Extremum
Consider \(f(x)=x^3\). Check \(x=0\) for an inflection point and for a local extremum.
The derivatives are
For \(x<0\), \(6x<0\), so the function is concave down. For \(x>0\), \(6x>0\), so it is concave up. The concavity changes across zero. Also, \(f(0)=0\), and \(f\) is continuous there. Hence \((0,0)\) is an inflection point.
To check for a local extremum, note that \(f'(x)=3x^2>0\) for every \(x\ne 0\). The function is increasing on both sides of zero; it does not switch from increasing to decreasing or from decreasing to increasing. For a direct value check, if \(x<0\), then \(x^3<0=f(0)\), while if \(x>0\), then \(x^3>0=f(0)\). There are nearby function values both below and above \(f(0)\), so zero is neither a local maximum nor a local minimum.
Finally, \(f'(0)=0\) and \(f''(0)=0\). These values do not make zero an extremum or inflection automatically; the interval sign checks establish the two conclusions separately.
An inflection point where \(f'(c)=0\), as in this example, is often called a stationary inflection point. The word “stationary” records that the tangent is horizontal; it does not change the requirement that concavity must change.
When the Second Derivative Is Undefined
If \(f''(c)\) is undefined, that point may still be an inflection point, but it may also fail to be one. The right procedure is the same: determine the concavity on the intervals to the left and right. First ensure that \(c\) belongs to the domain and that \(f\) is continuous there, so the graph contains the point \((c,f(c))\).
Worked Example: An Undefined Second Derivative Without an Inflection
Consider \(f(x)=x^{2/3}\), defined for all real \(x\). Determine whether the origin is an inflection point or a local extremum.
For \(x\ne0\), differentiate to obtain
The formula for \(f''\) is undefined at zero. For every \(x\ne0\), however, \(x^{-4/3}>0\), so
This inequality holds on both sides of zero. The graph is concave down on each side, so its concavity does not change. Since \(f(0)=0\) and \(f\) is continuous at zero, the origin lies on the graph, but it is not an inflection point.
It is a local minimum: \(x^{2/3}=|x|^{2/3}\ge 0=f(0)\) for every real \(x\). Thus this example also shows that a local extremum can occur at a point where the second derivative is undefined, without that point being an inflection point.
As an additional derivative check, \(f'(x)<0\) for \(x<0\) and \(f'(x)>0\) for \(x>0\). The function decreases toward zero from the left and increases to the right, consistent with the local minimum.
A Reliable Sign-Change Test
The sign test works because the second derivative controls how the first derivative changes. Here is the reasoning for the common case where \(f''\) is negative on one side and positive on the other. Assume \(f\) is twice differentiable on each of the two punctured intervals around \(c\); the derivatives need not exist at \(c\) itself.
If \(f''<0\) on the left interval and \(f''>0\) on the right interval, the second derivative has opposite signs on the two sides.
By the Mean Value Theorem applied to \(f'\) within either interval, \(f''<0\) makes \(f'\) strictly decreasing on the left, while \(f''>0\) makes \(f'\) strictly increasing on the right.
A decreasing derivative corresponds to concavity down; an increasing derivative corresponds to concavity up. Thus the graph changes concavity across \(c\).
Check that \(f(c)\) is defined and that \(f\) is continuous at \(c\). Then \((c,f(c))\) is an inflection point.
For completeness, the derivative-to-concavity step follows from slope comparisons. If \(f'\) is increasing on an interval, take three points \(a<b<d\) in that interval. By the Mean Value Theorem, the slope from \(a\) to \(b\) equals \(f'(u)\) for some \(u\in(a,b)\), and the slope from \(b\) to \(d\) equals \(f'(v)\) for some \(v\in(b,d)\). Since \(u<v\) and \(f'\) is increasing,
This ordering of successive slopes is the convex, or concave-up, behavior of the graph. If \(f'\) is decreasing, the inequality reverses and the graph is concave down. Applying this reasoning on each side of \(c\) justifies the sign-change test without requiring \(f''(c)\) to exist.
Do Not Confuse Inflection Points with Extrema
A local extremum is identified by a change in the direction of the function’s values; an inflection point is identified by a change in concavity. Checking one feature cannot substitute for checking the other. For example, the function \(x^3\) has an inflection point without an extremum, while \(x^{2/3}\) has a local minimum without an inflection point.
The following comparison summarizes what the examples show:
| Function at \(x=0\) | Second-derivative behavior | Concavity changes? | Local extremum? |
|---|---|---|---|
| \(x^4\) | \(f''(0)=0\); positive on both sides | No | Yes, a local minimum |
| \(x^3\) | Negative on the left, positive on the right | Yes | No |
| \(x^{2/3}\) | Undefined at zero; negative on both sides | No | Yes, a local minimum |
Notice that \(x^4\) has both a local minimum and a zero second derivative at zero, but still no inflection point. This is another reason not to treat \(f''(c)=0\) as a stand-alone classification rule.
Common Mistakes and Their Corrections
Mistake 1: “The second derivative is zero, so the point is an inflection.” For \(f(x)=x^4\), the incorrect reasoning is
The implication is invalid. In fact, \(f''(x)=12x^2>0\) on both sides of zero, so there is no concavity change. The correction is to compare the signs of \(f''\) on neighboring intervals.
Mistake 2: “The second derivative is undefined, so the point is an inflection.” For \(f(x)=x^{2/3}\), \(f''(0)\) is undefined, but \(f''(x)<0\) for every \(x\ne0\). Concavity is down on both sides. Undefinedness signals that a sign check is needed; it does not supply the result of that check.
Mistake 3: “An inflection point must be a local maximum or minimum.” For \(f(x)=x^3\), concavity changes from down to up, but the function increases on both sides of zero. A concavity change does not require the function’s direction of movement to reverse.
A Practical Checklist
When a point is proposed as an inflection point, use a short, deliberate check:
- Confirm that \(c\) is an interior point of the domain and that \(f(c)\) exists.
- Check continuity at \(c\), so the graph contains the point \((c,f(c))\).
- Find intervals immediately to the left and right where concavity can be assessed.
- Use \(f''\) signs where the second derivative exists, or another valid concavity test if needed.
- Compare the concavity on the two sides. Opposite concavity means an inflection; unchanged concavity means it is not one.
- Check for a local extremum separately by examining function behavior or the sign of \(f'\).
The checklist separates candidate-finding from classification. A zero or undefined second derivative can help locate where to investigate, but the final decision comes from the behavior around the point.
Check Your Understanding
For each question, explain which behavior on neighboring intervals settles the classification.
- For \(f(x)=x^6\), what is \(f''(0)\), and does concavity change at zero?
- For \(f(x)=x^3+x\), does the graph have an inflection point at zero? Is zero a local extremum?
- Why is an undefined value of \(f''(c)\) not enough to conclude that \(c\) is an inflection point?
- Can a function have an inflection point where \(f'(c)=0\)? Give the example from this tutorial and explain the sign change.
- What separate checks would you use to decide whether \(c\) is an inflection point and whether it is a local maximum or minimum?
