Concavity Describes Changing Tangent Slopes
A graph’s tangent line gives its instantaneous slope at a point. Concavity describes how those tangent slopes change as we move from left to right. When tangent slopes increase, the graph bends upward; when they decrease, it bends downward.
This description is separate from whether the function itself rises or falls. A function can be decreasing while its tangent slopes increase, or increasing while its tangent slopes decrease. To decide concavity, compare slopes at different points rather than looking only at whether the slopes are positive or negative.
Here, nondecreasing means that \(f'(x_1)\le f'(x_2)\) whenever \(x_1<x_2\) in the interval; nonincreasing means that \(f'(x_1)\ge f'(x_2)\). A tangent slope can be negative and still increase: for example, it can change from \(-5\) to \(-2\). What matters is the order of the slopes, not their signs.
Reading the Shape from Tangent Slopes
Imagine moving along a graph from left to right and recording the slope of the tangent at each point. If the recorded slopes grow, the graph becomes more tilted upward—or less tilted downward—as you move along it. That is concave-up behavior. If the recorded slopes shrink, the graph becomes less tilted upward—or more tilted downward. That is concave-down behavior.
For a concave-up graph, a sequence of tangent slopes might be \(-4,-1,2,5\). The graph initially falls, but its fall becomes gentler, and eventually it rises. For a concave-down graph, slopes such as \(5,2,-1,-4\) show the reverse pattern. In both cases, the function may rise or fall over some or all of the interval. Concavity tracks the change in slope, not the function’s direction of travel.
Worked Example: Tangent Slopes of a Parabola
Consider \(f(x)=x^2\) on the real line. Differentiate:
If \(x_1<x_2\), then multiplying that inequality by \(2\) gives \(2x_1<2x_2\). Thus \(f'(x_1)<f'(x_2)\): the tangent slopes strictly increase across the interval. The graph is concave up.
For a concrete comparison, the tangent slopes at \(x=-1,0,2\) are
These values increase from \(-2\) to \(0\) to \(4\). The function values at those points are \(f(-1)=1\), \(f(0)=0\), and \(f(2)=4\). The tangent lines at \(x=-1\) and \(x=2\) are, respectively,
Rearranging gives \(y=-2x-1\) and \(y=4x-4\), whose slopes are \(-2\) and \(4\), as required. As a check on the derivative calculation, differentiating \(x^2\) gives \(2x\), which produces both tangent slopes above.
Why Increasing Tangent Slopes Mean an Upward Bend
The definition links concavity directly to the behavior of \(f'\). The Mean Value Theorem helps explain why this change in tangent slopes also describes the shape of the curve between points. In particular, if \(f'\) is nondecreasing, then the average slope over a left-hand interval cannot exceed the average slope over the next interval.
To justify the claim, apply the Mean Value Theorem separately to \([a,b]\) and \([b,c]\). The theorem applies because differentiability on the interval ensures continuity on each closed subinterval and differentiability in its interior. It gives points \(u\in(a,b)\) and \(v\in(b,c)\) such that
Since \(u<b<v\), we have \(u<v\). If \(f'\) is nondecreasing, \(f'(u)\le f'(v)\). Substituting the two secant-slope equalities proves the stated inequality. If \(f'\) is nonincreasing, then \(f'(u)\ge f'(v)\), which gives the reversed inequality. Thus, in the concave-up case, average slopes over successive intervals move upward; in the concave-down case, they move downward.
For a differentiable function, the slope at \(x\) is \(f'(x)\).
Check whether \(f'\) is nondecreasing or nonincreasing on the interval of interest.
Nondecreasing tangent slopes mean concave up; nonincreasing tangent slopes mean concave down.
Concavity Does Not Tell You Whether the Function Rises
A positive derivative means tangent lines have positive slope, so the function is increasing. A negative derivative means tangent lines have negative slope, so the function is decreasing. Neither sign alone tells you whether those slopes are themselves increasing or decreasing. Concavity asks about comparisons between derivative values at different inputs.
Worked Example: A Decreasing Function That Is Concave Up
Consider \(f(x)=e^{-x}\) for all real \(x\). Its derivative is
For any \(x_1<x_2\), the exponential function gives \(e^{-x_1}>e^{-x_2}\), since \(-x_1> -x_2\). Multiplying by \(-1\) reverses the inequality, so
Therefore \(f'(x_1)<f'(x_2)\): the tangent slopes increase, and the graph is concave up. Yet every derivative value is negative, so the function is decreasing. For instance, the tangent slopes at \(x=0,1,2\) are \(-1,-1/e,-1/e^2\). Because \(1>1/e>1/e^2\), their negatives satisfy \(-1<-1/e<-1/e^2\), confirming the increase.
At \(x=0\), \(f(0)=1\), and the tangent line is \(y-1=-(x-0)\), or \(y=1-x\); its slope is \(-1=f'(0)\). At \(x=1\), \(f(1)=1/e\), and the tangent line is \(y-1/e=-(1/e)(x-1)\); its slope is \(-1/e=f'(1)\). These tangent-line checks agree with the derivative formula. Differentiating \(e^{-x}\) gives \(-e^{-x}\), as used throughout.
Concavity Can Depend on the Interval
A function does not have to be concave in the same direction throughout its entire domain. Its tangent slopes may decrease on one interval and increase on another. Therefore, state the interval whenever you describe concavity, and compare derivative values only within the interval being considered.
Worked Example: A Cubic with Different Slope Trends
Let \(f(x)=x^3\). Its derivative is
On the interval \(x<0\), as \(x\) increases toward \(0\), its square decreases. For example, the tangent slopes at \(-3,-2,-1\) are
They decrease from \(27\) to \(12\) to \(3\), so the graph is concave down on the negative side. At \(x=-2\), \(f(-2)=-8\), and the tangent line is \(y+8=12(x+2)\), or \(y=12x+16\). Its slope \(12\) agrees with \(f'(-2)\). At \(x=-1\), \(f(-1)=-1\), and the tangent line is \(y+1=3(x+1)\), or \(y=3x+2\); its slope \(3\) agrees with \(f'(-1)\).
On the interval \(x>0\), squares increase as \(x\) increases, so the tangent slopes increase. For instance, \(f'(1)=3\), \(f'(2)=12\), and \(f'(3)=27\), showing concave-up behavior on the positive side. At \(x=2\), \(f(2)=8\), and the tangent line is \(y-8=12(x-2)\), or \(y=12x-16\); its slope \(12\) matches \(f'(2)\). At \(x=1\), \(f(1)=1\), and the tangent line is \(y-1=3(x-1)\), or \(y=3x-2\); its slope \(3\) matches \(f'(1)\).
The slope trend is not the same on both sides: \(f'(-1)=3>f'(0)=0\), but \(f'(0)=0<f'(1)=3\). So \(f'\) is neither nondecreasing nor nonincreasing across the whole real line. This example illustrates why concavity descriptions must be tied to intervals.
A Common Mistake: Using the Sign of the Derivative
A common error is to decide concavity from whether \(f'\) is positive or negative. For \(f(x)=e^{-x}\), for example, one might write, “\(f'(x)=-e^{-x}<0\), so the graph is concave down.” That calculation only shows that the function is decreasing. It does not compare tangent slopes at different inputs.
The correction is to compare derivative values. In this example, \(f'(0)=-1\) and \(f'(1)=-1/e\). Since \(e>1\), we have \(1/e<1\), and therefore \(-1<-1/e\). The tangent slope has increased from \(x=0\) to \(x=1\), even though both slopes are negative. The graph is concave up on that interval.
Check Your Understanding
For each question, focus on how tangent slopes compare as the input increases.
- If the tangent slopes at successive points are \(-6,-3,0,4\), what concavity does this pattern indicate?
- A function has negative derivative values throughout an interval, and those values increase from left to right. Is the function increasing or decreasing? Is it concave up or concave down?
- Explain why a positive derivative at every point does not, by itself, determine concavity.
- For \(f(x)=x^3\), compare \(f'(-2)\), \(f'(-1)\), and \(f'(0)\). What slope trend do these values show?
- State the condition on \(f'\) that corresponds to a graph being concave down on an interval.
