Why the Direction of Error Matters
A tangent line gives a simple linear approximation to a function near a chosen point. But knowing that an approximation is close does not tell us whether it is too high or too low. When the function has a consistent curvature across an interval, concavity answers that question: it determines which side of the function its tangent line lies on.
This direction can matter in applications. A tangent-line estimate that is guaranteed to lie above a function can serve as an upper bound; one guaranteed to lie below can serve as a lower bound. The guarantee depends on the function’s behavior across the interval, not just at the point where the tangent is drawn.
Concavity, Convexity, and the Second Derivative
Let \(f\) be a function defined on an interval \(I\). The function is concave on \(I\) if, for any \(x,y\in I\) and \(0\leq\lambda\leq1\), its graph lies above or on the chord joining \((x,f(x))\) and \((y,f(y))\). In symbols, this means \(f(\lambda x+(1-\lambda)y)\geq \lambda f(x)+(1-\lambda)f(y)\). A function is convex if the inequality is reversed: its graph lies below or on each such chord.
For a twice-differentiable function on an interval, the sign of the second derivative is a convenient test: \(f''(x)\leq0\) throughout the interval implies concavity, while \(f''(x)\geq0\) throughout implies convexity. When these inequalities are strict throughout, the function is strictly concave or strictly convex, respectively. In geometric language, a concave graph bends down and a convex graph bends up.
The terms “concave down” and “convex up” are also commonly used for \(f''\leq0\) and \(f''\geq0\), respectively. The key point for estimating is to check the sign over the relevant interval. A second derivative with the right sign only at the tangent point does not, by itself, guarantee a one-sided estimate elsewhere.
The Tangent-Line Comparison
Here is a proof of the comparison for a function with nondecreasing derivative. The proof considers points on both sides of the tangent point because the sign of \(x-a\) changes.
For \(x\ne a\), the theorem on the interval between \(a\) and \(x\) gives a point \(\xi\) strictly between them such that \(f(x)-f(a)=f'(\xi)(x-a)\).
If \(x>a\), then \(\xi>a\). Since \(f'\) is nondecreasing, \(f'(\xi)\geq f'(a)\). Multiplying by the positive number \(x-a\) gives \(f'(\xi)(x-a)\geq f'(a)(x-a)\).
If \(x<a\), then \(\xi<a\), so \(f'(\xi)\leq f'(a)\). Now \(x-a<0\), so multiplying reverses the inequality: \(f'(\xi)(x-a)\geq f'(a)(x-a)\).
In either case, \(f(x)-f(a)\geq f'(a)(x-a)\), which is the claimed tangent-line bound. At \(x=a\), equality holds directly. Reversing the derivative comparisons proves the result for a nonincreasing derivative.
When \(f''\geq0\) throughout the interval, the Mean Value Theorem applied to \(f'\) shows that \(f'\) is nondecreasing there: for \(u<v\), it gives \(f'(v)-f'(u)=f''(\eta)(v-u)\geq0\) for some \(\eta\in(u,v)\). The corresponding argument with \(f''\leq0\) gives a nonincreasing derivative. This connects the second-derivative test to the tangent-line comparison.
Worked Example: A Concave Square-Root Function
Worked Example: Estimating a Square Root from Above
Use the tangent line to \(f(x)=\sqrt{x}\) at \(a=4\) to estimate \(\sqrt{9}\), and determine the direction of the error on \(x>0\).
First calculate the derivatives:
For every \(x>0\), \(f''(x)<0\), so the function is concave on this interval. At the tangent point, \(f(4)=2\) and \(f'(4)=\frac14\). Thus the linearization is
At \(x=9\), \(L(9)=2+\frac14(9-4)=2+\frac54=\frac{13}{4}=3.25\). The exact value is \(\sqrt{9}=3\). Therefore the tangent estimate is above the function, as predicted. The difference is \(\frac{13}{4}-3=\frac14\). Substitution back into the comparison gives \(f(9)=3\leq\frac{13}{4}=L(9)\), verifying the direction.
Worked Example: A Convex Exponential Function
Worked Example: Estimating an Exponential Value from Below
Use the tangent line to \(f(x)=e^x\) at \(a=0\) to estimate \(e^{0.2}\), and compare the estimate with the function value.
The derivatives are \(f'(x)=e^x\) and \(f''(x)=e^x\). Since \(e^x>0\) for every real \(x\), \(f\) is convex. At \(a=0\), \(f(0)=1\) and \(f'(0)=1\), so
At \(x=0.2\), the tangent estimate is \(L(0.2)=1+0.2=1.2\). The function value is \(e^{0.2}\approx1.221403\), rounded to six decimal places. Hence the estimate is below the function value, with difference approximately \(1.221403-1.2=0.021403\). Substituting into the comparison confirms \(1.2\leq e^{0.2}\), as expected for a convex function.
Worked Example: Checking Both Sides of a Tangent Point
Worked Example: A Parabola with an Exact Comparison
For \(f(x)=x^2\), find the tangent line at \(a=1\), then compare it with the function at \(x=0\) and \(x=2\).
We have \(f'(x)=2x\) and \(f''(x)=2>0\), so \(f\) is convex on the real line. At \(a=1\), \(f(1)=1\) and \(f'(1)=2\). The tangent line is
At \(x=0\), \(L(0)=-1\), while \(f(0)=0\); therefore \(L(0)=-1\leq0=f(0)\). At \(x=2\), \(L(2)=3\), while \(f(2)=4\); therefore \(L(2)=3\leq4=f(2)\). On both sides of the tangent point the tangent lies below the graph. The exact differences are \(f(0)-L(0)=1\) and \(f(2)-L(2)=1\), confirming the comparison in each case.
When the Direction Can Change
The tangent-line conclusion requires concavity or convexity on the interval being considered. If the second derivative changes sign, the graph may bend in different directions on opposite sides of the tangent point, and a single tangent line need not be consistently above or below the function.
For example, let \(f(x)=x^3\) and take \(a=0\). Since \(f(0)=0\) and \(f'(0)=0\), the tangent line is \(L(x)=0\). At \(x=1\), \(f(1)=1>L(1)\), but at \(x=-1\), \(f(-1)=-1<L(-1)\). The second derivative \(f''(x)=6x\) changes sign at zero, so there is no one-sided tangent bound across an interval containing both negative and positive values.
Using the Direction of Error
Concavity provides more than a description of a graph’s shape. It gives a reliable way to interpret a linearization. If \(f''\leq0\) across the interval, the tangent value is an upper bound for the function value. If \(f''\geq0\), the tangent value is a lower bound. The inequalities include equality at the tangent point, and may also include equality elsewhere if the function is linear on part of the interval.
This direction is distinct from the size of the error. Knowing that a tangent estimate is an upper or lower bound does not, by itself, say how far the estimate is from the true value. It does let you report which way the approximation errs, and it can help establish a one-sided bound without calculating the exact function value.
Before using the conclusion, check that the tangent point and the target input lie in an interval on which the required concavity or convexity holds. If the second derivative changes sign along the way, restrict the interval if possible or avoid claiming a direction that the available information does not guarantee.
Check Your Understanding
For each question, use the function’s behavior on the relevant interval, not merely at the tangent point.
- If \(f''(x)\leq0\) throughout an interval, which way does the tangent line at a point in that interval compare with the function?
- For a convex function, does the tangent-line approximation give a lower bound or an upper bound?
- Why must the sign of the second derivative be checked across the interval between the tangent point and the target input?
- For \(f(x)=x^2\), write the tangent line at \(a=1\) and check its position relative to the function at \(x=3\).
- What feature of \(f(x)=x^3\) prevents its tangent at zero from being a one-sided bound on both sides of zero?
