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Concave Function — Definition, Formula & Examples

A concave function is a function whose graph curves downward, meaning the line segment connecting any two points on the graph lies at or below the curve itself. It is also commonly called "concave down."

A function ff defined on an interval II is concave if for all x1,x2∈Ix_1, x_2 \in I and every λ∈[0,1]\lambda \in [0,1], the inequality f(λx1+(1−λ)x2)≥λf(x1)+(1−λ)f(x2)f(\lambda x_1 + (1-\lambda)x_2) \geq \lambda f(x_1) + (1-\lambda)f(x_2) holds. Equivalently, if ff is twice differentiable on II, then ff is concave if and only if f′′(x)≤0f''(x) \leq 0 for all x∈Ix \in I.

Key Formula

f(λx1+(1−λ)x2)≥λf(x1)+(1−λ)f(x2)f(\lambda x_1 + (1-\lambda)x_2) \geq \lambda f(x_1) + (1-\lambda)f(x_2)
Where:
  • ff = The function being tested for concavity
  • x1,x2x_1, x_2 = Any two points in the domain
  • λ\lambda = A weight between 0 and 1 that parametrizes the line segment

How It Works

To determine whether a function is concave, take its second derivative. If f′′(x)≤0f''(x) \leq 0 throughout the interval, the function is concave there. Geometrically, a concave function bends downward like an upside-down bowl. At any point on a concave function, the tangent line lies above the curve, which means the linear approximation always overestimates the function's value.

Worked Example

Problem: Show that f(x)=−x2+4xf(x) = -x^2 + 4x is a concave function on all of R\mathbb{R}.
Step 1: Find the first derivative.
f′(x)=−2x+4f'(x) = -2x + 4
Step 2: Find the second derivative.
f′′(x)=−2f''(x) = -2
Step 3: Since f′′(x)=−2<0f''(x) = -2 < 0 for all xx, the second derivative is negative everywhere.
Answer: Because f′′(x)<0f''(x) < 0 for all xx, the function f(x)=−x2+4xf(x) = -x^2 + 4x is concave on R\mathbb{R}.

Why It Matters

Concavity is central to optimization: a concave function on an interval has at most one maximum, and any local maximum is automatically the global maximum. This property is used extensively in economics (maximizing utility or profit) and in machine learning when analyzing loss functions.

Common Mistakes

Mistake: Confusing concave with convex. Some students think a function that opens downward is convex because it looks like a "dome."
Correction: A concave function curves downward (f′′≤0f'' \leq 0), while a convex function curves upward (f′′≥0f'' \geq 0). Remember: concave = cave = hollowed out on top.