Concavity of Sums and Scalar Multiples

Combining Functions Can Combine Their Concavity

A function is often built by adding simpler functions or multiplying one function by a constant. These operations change the function’s values, but they also have predictable effects on its shape. For sums, the effects reinforce one another when the functions bend in the same direction. For scalar multiplication, the sign of the scalar determines whether the bending is preserved or reversed.

The key is to keep track of what the concavity claim means and where it applies. Concavity is a property on an interval, and a sum is defined only where all its terms are defined. When the functions are twice differentiable, their second derivatives give a convenient way to check the rules. The defining inequality also shows directly why a negative scalar reverses concavity.

Concavity and Convexity

Let \(I\) be an interval and let \(f:I\to\mathbb R\). For \(x,y\in I\) and \(t\in[0,1]\), the point \(tx+(1-t)y\) lies between \(x\) and \(y\), so it belongs to \(I\). The function is concave on \(I\) when its value at this intermediate point is at least the corresponding weighted average of its endpoint values.

Geometrically, a concave function lies above each chord joining two points on its graph; a convex function lies below each chord. The inequalities allow equality, so a linear function is both concave and convex. This non-strict convention matters: adding or scaling functions may result in equality rather than strict bending.

For a function that is twice differentiable on an interval, the second-derivative criterion gives a practical test: if \(f''(x)\leq 0\) throughout the interval, then \(f\) is concave there; if \(f''(x)\geq 0\) throughout, then \(f\) is convex there. These are sufficient conditions under the stated differentiability assumption. In the usual calculus setting of a twice-differentiable function on an interval, the criterion also characterizes concavity and convexity. A sign that changes within the interval does not establish either property on the whole interval.

Domains of Sums and Scalar Multiples

If \(f\) is defined on an interval \(I\) and \(g\) on an interval \(J\), then \(f+g\) is defined on the common domain \(I\cap J\). The rules below apply on any interval contained in that common domain. For more than two functions, use the intersection of all their domains. A sum might therefore be concave on one interval but not on a larger domain.

For a real constant \(c\), the scalar multiple \(cf\) has the same domain as \(f\), since its values are obtained by multiplying \(f(x)\) by \(c\). The derivative rules for sums and constant multiples are \( (f+g)''=f''+g'' \) and \( (cf)''=cf'' \), when the functions are twice differentiable. The defining inequality proves the concavity rules without requiring differentiability.

Why Adding Concave Functions Preserves Concavity

Suppose \(f\) and \(g\) are concave on the same interval \(I\). Fix any \(x,y\in I\) and \(t\in[0,1]\). By concavity, the defining inequality holds for \(f\) and also for \(g\). Adding those two inequalities gives the inequality for their sum:

Because \(x,y,t\) were arbitrary, this proves that \(f+g\) is concave on \(I\). The same argument, with both inequalities reversed, proves that the sum of two convex functions is convex.

The second-derivative calculation gives the corresponding calculus check: if \(f''\leq 0\) and \(g''\leq 0\) throughout \(I\), then \((f+g)''=f''+g''\leq 0\). This agrees with the defining-inequality proof. It is important that both component functions have the required sign; if one is concave and the other convex, their sum’s concavity cannot be decided from those labels alone.

Why the Sign of a Scalar Matters

For \(c>0\), multiply the defining inequality for a concave function \(f\) by \(c\). Since multiplication by a positive number preserves inequality direction, the result is exactly the concavity inequality for \(cf\). If \(c=0\), then \(cf\) is the constant zero function, which is both concave and convex.

For \(c<0\), multiplication reverses the inequality direction. Thus, if \(f\) is concave, the result is the defining inequality for \(cf\) to be convex. In second-derivative terms, \( (cf)''=cf'' \); multiplying a nonpositive second derivative by a negative constant makes it nonnegative. Both arguments explain the reversal.

More generally, a finite sum \(\sum_{i=1}^{n}c_i f_i\) is concave if each \(f_i\) is concave and every coefficient \(c_i\) is nonnegative. Each term \(c_i f_i\) is then concave, and repeated application of the sum rule gives the result. Negative coefficients require extra care: they turn concave terms into convex ones, so the resulting sum needs to be checked rather than classified from the components alone.

Worked Example: Adding Two Concave Quadratics

Worked Example: A Negative Scalar Reverses Concavity

When Components Bend in Different Directions

Having one concave summand and one convex summand does not determine the concavity of the sum. Their second derivatives combine numerically, and the result may be negative, positive, or zero. The answer depends on their relative sizes. This is not an exception to the addition rule: that rule guarantees preservation when the summands bend in the same required direction.

A Common Mistake: Forgetting That Negative Multiplication Reverses Inequalities

A frequent error is to say that multiplying by any constant preserves concavity. Consider the concave function \(f(x)=-x^2\) and \(q(x)=-f(x)\). An incorrect argument might start from \(f''(x)=-2\) and claim that multiplying by \(-1\) still gives a nonpositive second derivative. But the calculation is

The correction is that the sign of the multiplier matters: \(q''=(-1)f''=(-1)(-2)=2\), so \(q\) is convex, not concave. The same reversal appears in the defining inequality. Multiplying an inequality by a negative number reverses its direction; keeping the direction unchanged is the error.

A Reliable Way to Classify a Combination

When functions are combined, first identify an interval on which all terms are defined. If the signs of the summands’ concavity agree and all coefficients are nonnegative, the preservation rules settle the answer immediately. Otherwise, when the terms are twice differentiable, compute the second derivative of the whole expression and determine its sign on the interval. A single sign at one point does not establish concavity or convexity on an entire interval.

These rules matter because complicated functions often consist of simpler pieces. Knowing how addition and scalar multiplication affect concavity lets you recognize when the shape follows immediately from the pieces—and when it must be checked anew. The essential distinction is between combining functions that bend consistently and combining contributions that may compete.