Why a Zero Second Derivative Is Not the Definition
Concavity describes how a graph bends. An inflection point marks a place where that bending changes direction: the graph changes from concave up to concave down, or from concave down to concave up.
The second derivative is useful for describing concavity where it exists. But the defining feature of an inflection point is the change in concavity itself. A value of \(f''\) equal to zero at one input does not, by itself, tell us how the graph bends on either side.
Definition: Change in Concavity
Use the standard chord-based meanings of concavity: a function is concave up on an interval when its graph lies below the chords joining points of the graph; it is concave down when its graph lies above those chords. When a function is twice differentiable on an interval, the second-derivative criterion gives a convenient way to determine these behaviors: \(f''>0\) implies concavity up, and \(f''<0\) implies concavity down.
The definition concerns behavior on both sides of \(c\). The function must be defined at \(c\), so that \((c,f(c))\) is actually a point on its graph. Continuity at \(c\) ensures the graph joins at that point rather than breaking there.
This definition does not require \(f''(c)\) to exist. It also does not say that every input where \(f''(c)=0\) is an inflection point. Both observations follow from focusing on the change in concavity rather than on a single derivative value.
How Second-Derivative Signs Show a Change
The second-derivative criterion connects the sign of \(f''\) on an interval to concavity there. Applying it on each side of \(c\) gives a direct justification for a common way of recognizing a change in concavity.
If \(f''(x)>0\) for inputs just left of \(c\), the second-derivative criterion says the function is concave up there. If \(f''(x)<0\), it is concave down there.
Apply the same criterion on an interval just right of \(c\), where \(f\) is twice differentiable.
If the signs give opposite concavities on the two sides, and \(f\) is defined and continuous at \(c\), the graph changes concavity at \((c,f(c))\).
In particular, if \(f''\) is negative just to the left of \(c\) and positive just to the right, the function changes from concave down to concave up. If the signs are positive and then negative, it changes from concave up to concave down. These conclusions require the stated signs on intervals around \(c\), not merely a value at \(c\).
Worked Example: A Cubic That Changes from Down to Up
Worked Example: A Cubic That Changes from Down to Up
Consider \(f(x)=x^3\). Find its concavity on either side of \(x=0\), and identify the inflection point if there is one.
Differentiate once:
Differentiate again:
For \(x<0\), \(6x<0\), so the second-derivative criterion shows that \(f\) is concave down to the left of zero. For \(x>0\), \(6x>0\), so it is concave up to the right. Thus the concavity changes at \(c=0\).
The function is defined and continuous there, and its graph point is
Therefore \((0,0)\) is an inflection point. At the input itself, \(f''(0)=6(0)=0\); the inflection conclusion, however, follows from the opposite concavities on the two sides. As a check, substituting \(x=0\) into \(f(x)=x^3\) gives \(f(0)=0\), the stated vertical coordinate.
Worked Example: A Zero Second Derivative Without a Change
Worked Example: A Zero Second Derivative Without a Change
Consider \(g(x)=x^4\). The second derivative is zero at the origin. Determine whether that point is an inflection point.
Differentiate:
For every \(x\), \(12x^2\geq 0\). In particular, \(g''(x)>0\) for every \(x\neq 0\). The second-derivative criterion shows that \(g\) is concave up on both sides of zero. Its concavity therefore does not change there.
At the input \(x=0\),
Although \((0,0)\) lies on the graph and \(g''(0)=0\), it is not an inflection point because the function is concave up on both sides. As a direct check on the concavity sign, \(g''(-1)=12\) and \(g''(1)=12\), both positive.
This example isolates the distinction: \(f''(c)=0\) is a statement about the second derivative at one input; an inflection point requires opposite concavities on the two sides.
Worked Example: A Change from Up to Down
Worked Example: A Change from Up to Down
Consider \(h(x)=-x^3+3x^2\). Determine whether its graph has an inflection point at \(x=1\).
Differentiate once:
Differentiate again and factor:
If \(x<1\), then \(1-x>0\), so \(h''(x)>0\) and \(h\) is concave up. If \(x>1\), then \(1-x<0\), so \(h''(x)<0\) and \(h\) is concave down. The concavity changes at \(x=1\).
The function is a polynomial, so it is defined and continuous at \(1\). Evaluate its function value:
Thus the inflection point is \((1,2)\). The coordinate check is \(h(1)=2\), as required for a point on the graph. Also, \(h''(1)=6(1-1)=0\); again, the change in concavity, not just this zero, justifies the classification.
Worked Example: Concavity on Opposite Sides of a Missing Point
Worked Example: Concavity on Opposite Sides of a Missing Point
Consider \(q(x)=1/x\), whose domain is \(\mathbb{R}\setminus\{0\}\). The second derivative has opposite signs on the two sides of zero. Does that make zero an inflection point?
Differentiate:
For \(x<0\), \(x^3<0\), so \(q''(x)<0\) and the graph is concave down. For \(x>0\), \(x^3>0\), so \(q''(x)>0\) and the graph is concave up. These are opposite concavities.
However, \(q(0)\) is not defined. Consequently, there is no graph point \((0,q(0))\), and the function cannot be continuous there. The opposite concavities occur on either side of a domain break, not at an inflection point of \(q\). The sign change alone is not enough.
Common Mistake: Treating a Condition as a Definition
A tempting but incorrect shortcut is to define inflection points as inputs where the second derivative is zero. The \(x^4\) example disproves that shortcut: \(g''(0)=0\), but the function remains concave up on both sides. A zero is not enough.
The reverse shortcut is also unreliable: requiring \(f''(c)=0\) as part of the definition excludes any genuine change in concavity for which the second derivative does not exist at \(c\). The definition asks about the function’s behavior on the two sides and whether the point belongs to its graph.
Why the Definition Matters
An inflection point describes a qualitative change in the shape of a graph. It identifies a transition in how the graph bends, which is different from identifying a horizontal tangent, a local maximum, or a local minimum. Those features may occur at the same input in some functions, but none is the definition of an inflection point.
Keeping the definition separate from a computational clue prevents two errors: labeling a point merely because \(f''\) vanishes there, and labeling a missing domain point because the concavity differs on its two sides. In each case, the question to return to is the same: does the function change concavity at a point that is actually on its graph?
Check Your Understanding
For each question, focus on concavity on both sides and on whether the function is defined at the point.
- State the definition of an inflection point in terms of concavity and the graph.
- For \(f(x)=x^3\), what are the concavities immediately to the left and right of zero, and what is the inflection point?
- For \(g(x)=x^4\), why does \(g''(0)=0\) fail to make \((0,0)\) an inflection point?
- What conditions make a second-derivative sign change useful evidence of an inflection point?
- Why is zero not an inflection point of \(q(x)=1/x\), even though \(q''\) has opposite signs on the two sides of zero?
