Convexity and Concavity¶

Definition¶

Let $E$ be an interval and let $f\colon E\to\mathbb{R}$ be a function. The function $f$ is convex on $E$ if, for all $x_1,x_2\in E$ and every $0<\alpha<1\text{,}$

and concave on $E$ if the inequality is reversed. If $x_1\neq x_2$ always gives a strict inequality, we say that $f$ is strictly convex or strictly concave.

The point $(1-\alpha)x_1+\alpha x_2$ lies between $x_1$ and $x_2\text{,}$ which is why the domain has to be an interval.

Interpretation¶

Convexity compares a function with its chords: $f$ is convex when the chord between any two points of its graph lies on or above the graph, and concave when it lies on or below it. A convex graph bends upwards, a concave one downwards.

The dashed chords join the points of each graph at $x=-1.5$ and $x=1.5$$1.5$. For $y=x^2$ the chord lies at height $2.25$, while the graph dips to $0$ in the middle, so $x^2$ is convex. For $y=-x^2$ everything is reflected: the chord lies below the graph, and $-x^2$ is concave.

In the definition, the point $(1-\alpha)x_1+\alpha x_2$ runs from $x_1$ to $x_2$ as $\alpha$ runs from $0$ to $1$, and $(1-\alpha)f(x_1)+\alpha f(x_2)$ is the height of the chord above that point, with the same weights. So the inequality says: the value at a weighted average of two inputs is at most the same weighted average of the two values. For differentiable functions this is equivalent to $f'$ being increasing, and for twice differentiable functions to $f''\ge0$$0$, as the Key Results state; but convexity itself needs no derivatives, as $|x|$ shows.

Convexity makes optimization simple. A convex differentiable function lies above each of its tangent lines, so on an interval every stationary point of it is a global minimum. Some books say "concave up" for convex and "concave down" for concave; the meaning is the same. The averaging inequality returns in Probability Distributions as Jensen's inequality, and Math IV extends the definition to functions of several variables.

Key Results¶

• Tangent/secant line test. For a convex graph, secants lie above the curve and tangents lie below it; for a concave graph both comparisons reverse.
• A differentiable function $f$ is (strictly) convex on $E$ exactly when $f'$ is (strictly) increasing on $E\text{,}$ and (strictly) concave exactly when $f'$ is (strictly) decreasing.
• $f$ is (strictly) convex if and only if $-f$ is (strictly) concave.
• Necessary condition. If $f$ is twice differentiable at $x_0$ and convex on some interval $(x_0-\varepsilon,x_0+\varepsilon)\text{,}$ then $f''(x_0)\geq 0\text{;}$ if it is concave there, then $f''(x_0)\leq 0\text{.}$
• Sufficient condition. If $f''>0$ on $(a,b)$ then $f$ is strictly convex there; if $f''<0$ then strictly concave; and $f$ is affine on $(a,b)\text{,}$ i.e. of the form $f(x)=mx+c\text{,}$ if and only if $f''\equiv 0\text{.}$
• Complete criterion. For twice-differentiable $f\colon (a,b)\to\mathbb{R}\text{,}$ $f$ is convex on $(a,b)$ if and only if $f''\geq 0$ throughout, and concave if and only if $f''\leq 0$ throughout.
• Strict inequalities are sufficient for strictness but not necessary: $f(x)=x^4$ is strictly convex although $f''(0)=0\text{.}$

Examples¶

• The function $f(x)=\mathrm{e}^x$ is strictly convex on $\mathbb{R}$ since $f''(x)=\mathrm{e}^x>0\text{.}$
• The function $f(x)=-x^2$ is strictly concave on $\mathbb{R}\text{.}$
• The function $f(x)=|x|$ is convex although it is not differentiable at $x=0\text{;}$ it is not strictly convex.
• The function $f(x)=1/x^2$ is strictly convex on $(0,+\infty)$ and on $(-\infty,0)$ separately, but not on $(-\infty,0)\cup(0,+\infty)\text{,}$ which is not an interval.

Prerequisites¶

• Function
• Interval

Related Concepts¶

• Second-Order Condition
• Local Extremum
• Graph of a Function
• Higher-Order Derivative

Sources¶

• Math I: Calculus · Lecture 9

Aliases: concave up, concave down Keywords: convex function, concave function Symbols: alpha, element of, epsilon, infinity, real numbers, union

Last updated: 22 October 2026 19:04