From the Shape of \(f'\) to the Shape of \(f\)
Concavity describes how the slopes of a function change. The derivative \(f'\) records those slopes, so its behavior can reveal the concavity of \(f\). In particular, when \(f'\) changes from decreasing to increasing, \(f\) changes from concave down to concave up. If that change occurs at \(c\), then \(f\) has an inflection point at \(c\), provided \(f\) is defined and continuous there.
This connection gives another way to locate possible inflection points: study local maxima and minima of \(f'\). But the qualification matters. A local extremum of \(f'\) alone does not always guarantee that \(f\) changes concavity. We will establish the reliable version of the connection, then see why the qualification is necessary.
Why the Derivative's Monotonicity Matters
On an interval where \(f'\) is decreasing, the slopes of \(f\) decrease as \(x\) increases, so \(f\) is concave down there. Where \(f'\) is increasing, the slopes increase and \(f\) is concave up. When \(f''\) exists, the second-derivative criterion expresses the same relationship: \(f''<0\) indicates concavity down, and \(f''>0\) indicates concavity up.
The relationship between increasing \(f'\) and concavity can also be justified directly. For \(x<y<z\), the Mean Value Theorem gives points \(u\in(x,y)\) and \(v\in(y,z)\) such that
Since \(u<v\), if \(f'\) is increasing then \(f'(u)\leq f'(v)\). Thus the slope of the secant from \(x\) to \(y\) is no greater than the slope from \(y\) to \(z\), which is the slope-based condition for \(f\) to be convex (concave up). If \(f'\) is decreasing, the inequalities reverse, giving concavity down.
The hypotheses describe the behavior on intervals beside \(c\), not merely the value of \(f'\) at \(c\). A local minimum of \(f'\) tells us that nearby values of \(f'\) are no smaller than \(f'(c)\). It does not, by itself, say that \(f'\) decreases throughout the left-hand interval and increases throughout the right-hand interval. For polynomials, however, a strict local extremum of \(f'\) does produce those monotonic directions sufficiently close to the extremum. The examples below illustrate this useful setting.
Worked Polynomial Examples
Worked Example: A Local Minimum of \(f'\)
Let \(f(x)=x^3-3x+2\). Find the local extremum of \(f'\), determine the concavity on either side, and identify any resulting inflection point.
Differentiate \(f\):
This derivative is a quadratic with positive leading coefficient. Its vertex is at \(x=0\), so \(f'\) has a local minimum there. Differentiate again:
For \(x<0\), \(6x<0\), so \(f\) is concave down. For \(x>0\), \(6x>0\), so \(f\) is concave up. The concavity changes at \(0\). The point on the graph is
Thus \((0,2)\) is an inflection point. To check the computations, differentiating \(x^3-3x+2\) gives \(3x^2-3\), and differentiating that gives \(6x\), whose signs on either side of \(0\) confirm the concavity change.
Worked Example: A Local Maximum of \(f'\)
Let \(f(x)=-(x-2)^3+1\). Determine whether the local extremum of \(f'\) gives an inflection point.
Differentiating gives
Since \((x-2)^2\geq 0\), we have \(f'(x)\leq 0\), with equality at \(x=2\). The derivative therefore has a local maximum at \(2\). The second derivative is
If \(x<2\), then \(x-2<0\), so \(f''(x)>0\): \(f\) is concave up. If \(x>2\), then \(x-2>0\), so \(f''(x)<0\): \(f\) is concave down. The concavity changes from up to down. The function value at the change is
Therefore \((2,1)\) is an inflection point. Differentiating the proposed first derivative gives \(-6(x-2)\), as required, and its signs on the two sides verify the result.
Worked Example: Two Local Extrema of \(f'\)
Let \(f(x)=x^4-2x^2\). Find the local extrema of \(f'\), classify the concavity around them, and give the inflection points.
First compute both derivatives:
The local extrema of \(f'\) can occur where \(f''(x)=0\). Solving,
To determine the sign of \(f''\), factor it:
When \(|x|>\frac{1}{\sqrt{3}}\), \(3x^2-1>0\), so \(f\) is concave up. When \(|x|<\frac{1}{\sqrt{3}}\), \(3x^2-1<0\), so \(f\) is concave down. Consequently, \(f'\) changes from increasing to decreasing at \(-\frac{1}{\sqrt{3}}\), giving a local maximum, and from decreasing to increasing at \(\frac{1}{\sqrt{3}}\), giving a local minimum. The concavity of \(f\) changes at both points.
At either value \(x=\pm\frac{1}{\sqrt{3}}\), we have \(x^2=\frac13\) and \(x^4=\frac19\). Hence
The inflection points are \(\left(-\frac{1}{\sqrt{3}},-\frac59\right)\) and \(\left(\frac{1}{\sqrt{3}},-\frac59\right)\). As a check, the computed second derivative has exactly the signs used above: positive outside the two values and negative between them.
Why the Polynomial Connection Works
For a nonconstant polynomial \(f'\), a strict local minimum or maximum forces \(f''\) to change sign nearby. Here is a justification. Let \(P=f'\) and suppose \(P\) has a strict local minimum at \(c\). Since \(P\) is a polynomial, \(P(x)-P(c)\) has a first nonzero term in its expansion in powers of \(x-c\):
For a local minimum, this difference must be positive for all sufficiently close \(x\ne c\). That requires \(m\) to be even and \(a_m>0\): an odd power changes sign across \(c\), while an even power with a negative coefficient is negative on both sides. Differentiating the polynomial expansion gives
Because \(m-1\) is odd and \(m a_m>0\), the leading term is negative just to the left of \(c\) and positive just to the right. The higher-degree terms do not change those signs sufficiently close to \(c\). Thus \(P'=f''\) changes from negative to positive, and \(f\) changes from concave down to concave up. For a strict local maximum, the same argument has \(a_m<0\), so the signs reverse. This proves the polynomial connection for strict local extrema.
A Pitfall: A Local Extremum Alone Is Not Enough
A tempting but incorrect argument is: “\(f'\) has a local minimum at \(c\), so \(f'\) decreases to \(c\) and increases after \(c\); therefore \(f\) changes concavity there.” The first implication is not valid for every differentiable function. A local minimum compares the values of \(f'\) with \(f'(c)\); it does not guarantee that the values move monotonically as \(x\) approaches or leaves \(c\).
For a concrete example, define
Since \(1\leq 2+\sin(1/x^2)\leq 3\), we have \(g(x)>0\) for \(x\ne0\), so \(g\) has a strict local minimum at \(0\). By the Fundamental Theorem of Calculus, \(f'=g\). Also, \(g\) is differentiable at \(0\), since
For \(x\ne0\), however, differentiating gives
The term \(-2x\cos(1/x^2)\) makes \(f''\) take both positive and negative values arbitrarily close to \(0\) on each side. Thus there is no interval immediately to the left or right on which \(f\) has one consistent concavity, despite the local minimum of \(f'\). The correction is to check the actual monotonicity of \(f'\) on the two sides, or, when available, check that \(f''\) changes sign. For polynomials with strict local extrema of \(f'\), the preceding argument supplies that sign change.
Using the Connection Carefully
When investigating a possible inflection point suggested by a local extremum of \(f'\), organize the reasoning around the behavior on either side of the candidate:
Find a local maximum or minimum of \(f'\), or a point where \(f''\) may change sign.
Determine whether \(f'\) decreases then increases, or increases then decreases. If \(f''\) exists, its signs on the two sides provide a direct check.
A change in the sign of \(f''\), or the corresponding change in monotonicity of \(f'\), verifies the change in concavity.
Evaluate \(f(c)\). If \(f\) is continuous at \(c\), report the inflection point as \((c,f(c))\).
This approach is useful because it connects two descriptions of the same shape: the way slopes change and the way the graph bends. It also prevents a common error in classification: naming an inflection point from a single value such as \(f''(c)=0\), or from a local extremum of \(f'\), without confirming what happens on both sides.
Check Your Understanding
For each question, distinguish the information about a candidate point from the evidence needed to confirm a change in concavity.
- If \(f'\) decreases immediately to the left of \(c\) and increases immediately to the right, what concavity change does this indicate for \(f\)?
- For \(f(x)=x^3-6x^2+9x\), compute \(f'\) and \(f''\), then determine whether \(f'\) has a local extremum at \(x=2\) and whether \(f\) changes concavity there.
- Why does \(f''(c)=0\) alone not establish that \(c\) is an inflection point?
- What additional sign information about \(f''\) would confirm an inflection point at \(c\), when \(f''\) exists on either side?
- Why does a strict local extremum of a nonconstant polynomial \(f'\) force \(f''\) to change sign nearby?
