From the Second Derivative to Concavity
A function can rise or fall while bending in different ways. To describe that bending, we use concavity. The second derivative provides a practical test: where it is positive, the first derivative is increasing; where it is negative, the first derivative is decreasing.
This tutorial focuses on classifying concavity on intervals from the sign of \(f''\). The test must be applied on an interval where the function is twice differentiable, and the sign must hold throughout the interval—not merely at a selected point.
The weak conditions allow equality at some or even all points. In this convention, a function that is both concave up and concave down on an interval is a straight line there. The distinction between strict and weak concavity matters when the second derivative is zero at some points.
What the Sign Tells You
The first derivative describes how the function changes. The second derivative describes how the first derivative changes. In particular, a positive second derivative means that \(f'\) is increasing, while a negative second derivative means that \(f'\) is decreasing. This change in slope corresponds to the function bending upward or downward.
The criterion is about the sign throughout an interval. For example, knowing that \(f''(2)>0\) tells you something locally about the curvature near \(2\), but by itself it does not establish concavity on a whole interval. To classify a specified interval, check that the required sign holds at every point in it.
| Second-derivative condition on \(I\) | Conclusion on \(I\) |
|---|---|
| \(f''(x)>0\) throughout \(I\) | Strictly concave up |
| \(f''(x)\geq 0\) throughout \(I\) | Concave up |
| \(f''(x)<0\) throughout \(I\) | Strictly concave down |
| \(f''(x)\leq 0\) throughout \(I\) | Concave down |
| \(f''(x)=0\) throughout \(I\) | Affine; both concave up and concave down |
The terminology “concave up” is also commonly called “convex.” “Concave down” is also called “concave.” The labels refer to the shape, not to whether the function itself is positive or negative.
Worked Example: A Positive Second Derivative
Worked Example: A Positive Second Derivative
Classify the concavity of \(f(x)=x^2+2x+3\) on \(\mathbb{R}\).
First differentiate:
Differentiate again:
For every real \(x\), \(2>0\). Thus \(f''(x)>0\) throughout \(\mathbb{R}\), so the second-derivative criterion shows that \(f\) is strictly concave up on \(\mathbb{R}\).
As a check, \(f'\) is increasing: if \(a<b\), then
This agrees with the positive second derivative and the concavity classification.
Worked Example: A Negative Second Derivative
Worked Example: A Negative Second Derivative
Classify the concavity of \(g(x)=\ln x\) on its domain.
The domain is \((0,\infty)\). On this interval,
For every \(x>0\), \(x^2>0\), so \(-1/x^2<0\). Therefore \(g''\) is negative throughout its domain, and \(g\) is strictly concave down on \((0,\infty)\).
We can check the sign by testing any domain value, such as \(x=2\):
That substitution illustrates the sign, but the conclusion for the entire domain comes from the expression \(-1/x^2<0\) for every \(x>0\), not from the single test value.
When the Second Derivative Is Identically Zero
There is an important difference between \(f''\) being zero at a particular input and \(f''\) being zero at every point of an interval. If \(f''\) is identically zero on an interval, then \(f'\) is constant there and \(f\) is a straight line on that interval.
Here is a proof. Take any \(a,b\in I\) with \(a<b\). Since \(f\) is twice differentiable, \(f'\) is continuous on \([a,b]\) and differentiable on \((a,b)\). The Mean Value Theorem applied to \(f'\) gives a point \(c\in(a,b)\) such that
By assumption, \(f''(c)=0\). Hence \(f'(b)-f'(a)=0\), so \(f'(b)=f'(a)\). Because \(a\) and \(b\) were arbitrary, \(f'\) has one constant value, say \(m\), throughout \(I\). Apply the Mean Value Theorem to \(f\) between any two points \(a<b\) in \(I\). For some \(d\in(a,b)\),
Rearranging gives \(f(b)-mb=f(a)-ma\). Thus \(f(x)-mx\) has the same value, say \(b_0\), at every point in \(I\). Therefore \(f(x)=mx+b_0\) on \(I\), as claimed.
An affine function has constant slope, so its graph neither bends upward nor downward. It satisfies both weak concavity conditions. It is not strictly concave up or strictly concave down.
Worked Example: An Identically Zero Second Derivative
Worked Example: An Identically Zero Second Derivative
Classify the concavity of \(h(x)=4-3x\) on \(\mathbb{R}\).
Differentiate once and then twice:
Because \(h''(x)=0\) for every real \(x\), the theorem shows that \(h\) is affine. In fact, its given formula is already of the form \(mx+b\), with \(m=-3\) and \(b=4\).
Consequently, \(h\) is both concave up and concave down in the weak sense on \(\mathbb{R}\), but it is strictly neither. To check the calculation, differentiating the antiderivative steps in reverse gives \(\frac{d}{dx}(-3) = 0\), as required for \(h''\).
Concavity Can Differ on Different Intervals
A function need not have the same concavity everywhere. If the second derivative has different signs on different intervals, apply the criterion separately on each interval. This is why the conclusion should always name the interval on which it holds.
Worked Example: Classifying on Two Intervals
Let \(p(x)=x^3\). Determine its concavity on \((-\infty,0)\) and \((0,\infty)\).
Differentiate twice:
If \(x<0\), then \(6x<0\). Thus \(p''\) is negative throughout \((-\infty,0)\), and \(p\) is strictly concave down there. If \(x>0\), then \(6x>0\). Thus \(p''\) is positive throughout \((0,\infty)\), and \(p\) is strictly concave up there.
The endpoint \(0\) is not in either of those open intervals. The classification above concerns the sign of \(p''\) on each named interval; the conclusion for one interval should not be extended to the other.
For a direct sign check, choose one value in each interval. At \(x=-1\), \(p''(-1)=6(-1)=-6<0\). At \(x=1\), \(p''(1)=6(1)=6>0\). These checks agree with the expression \(6x\), whose sign is negative for every \(x<0\) and positive for every \(x>0\).
A Common Mistake: Using the First Derivative Instead
The first derivative and the second derivative answer different questions. The sign of \(f'\) describes whether \(f\) is increasing or decreasing; the sign of \(f''\) describes whether its slope is increasing or decreasing. Confusing these roles can produce an incorrect concavity conclusion.
For \(p(x)=x^3\), a mistaken argument might say:
The implication is wrong: \(p'\geq 0\) indicates that \(p\) is increasing, not that it is concave up. The correction is to calculate the second derivative:
Its sign is negative for \(x<0\) and positive for \(x>0\). Therefore \(p\) is concave down on \((-\infty,0)\) and concave up on \((0,\infty)\). Keeping the jobs of \(f'\) and \(f''\) separate prevents this error.
A Reliable Classification Routine
For a given function and interval, the process is short, but each part matters:
Find \(f'\), then calculate \(f''\) carefully.
Make sure the function is twice differentiable throughout that interval.
Use the formula for \(f''\) and the conditions defining the interval.
Positive means concave up; negative means concave down. Include the interval and whether the claim is strict or weak.
If \(f''\) is identically zero on the interval, state that the function is affine there and hence both weakly concave up and weakly concave down. If the sign is not the same throughout the interval, do not use a single positive-or-negative conclusion for that whole interval; examine the intervals on which the sign is consistent.
Why This Classification Matters
Concavity describes how a graph bends, but it also summarizes how its slope changes. When \(f''>0\), the slopes increase as \(x\) increases; when \(f''<0\), they decrease. This information helps organize the global shape of a function and complements what the first derivative tells you about increasing and decreasing behavior.
The most dependable habit is to write down the second derivative, establish its sign on the interval, and then state the conclusion precisely. A sign at one sample point is not a substitute for a sign argument on the whole interval, and the sign of the first derivative is not a concavity test.
Check Your Understanding
For each question, identify the interval or condition being discussed and justify your conclusion using the second derivative.
- If \(f''(x)=5\) for every \(x\in I\), what type of concavity does \(f\) have on \(I\)?
- If \(g''(x)=-2/x^2\) for every \(x>0\), classify the concavity of \(g\) on \((0,\infty)\).
- What can you conclude if \(h''(x)=0\) for every \(x\) in an interval \(I\)?
- For \(q(x)=x^3\), calculate \(q''(x)\) and classify concavity on \((-\infty,0)\) and \((0,\infty)\).
- Why does \(f'(x)\geq 0\) on an interval not by itself show that \(f\) is concave up there?
