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Convexity and Concavity of Functions

Basic Idea of Convexity

Convexity describes the bending direction of a function’s graph—an important property that reflects how the function changes across intervals.

Concave and Convex Functions

Convention note: different textbooks use “concave” and “convex” in opposite ways. Here we follow the common Chinese convention: a concave function bends downward (like

\ln x

), and a convex function bends upward (like

x^2

). When reading other books, always check their convention.

Concave Function

Let $f(x)$ be continuous on an interval $I$ . If for every $x_1, x_2 \in I$ and every $\lambda \in [0,1]$ ,

f(\lambda x_1 + (1-\lambda)x_2) \geq \lambda f(x_1) + (1-\lambda)f(x_2)

then $f(x)$ is said to be concave on $I$ .

Symbol	Type	Reading/Explanation	Meaning in This Article
$\lambda$	Greek letter	Lambda	Parameter used for convex combinations

Geometric interpretation

The graph bends downward; the chord between any two points lies below the graph.

Convex Function

Let $f(x)$ be continuous on an interval $I$ . If for every $x_1, x_2 \in I$ and every $\lambda \in [0,1]$ ,

f(\lambda x_1 + (1-\lambda)x_2) \leq \lambda f(x_1) + (1-\lambda)f(x_2)

then $f(x)$ is said to be convex on $I$ .

Geometric interpretation

The graph bends upward; the chord between any two points lies above the graph.

Tests for Convexity

Second-Derivative Test

Theorem1

Let $f(x)$ be twice differentiable on an interval $I$ :

If $f''(x) > 0$ on $I$ , then $f(x)$ is convex on $I$
If $f''(x) < 0$ on $I$ , then $f(x)$ is concave on $I$

First-Derivative Test

Theorem2

Let $f(x)$ be differentiable on an interval $I$ :

If $f'(x)$ is monotonically increasing on $I$ , then $f(x)$ is convex on $I$
If $f'(x)$ is monotonically decreasing on $I$ , then $f(x)$ is concave on $I$

Convexity of Common Functions

Quadratic Functions

For $f(x) = ax^2 + bx + c$ ( $a \neq 0$ ):

When $a > 0$ , $f''(x) = 2a > 0$ , so the function is convex
When $a < 0$ , $f''(x) = 2a < 0$ , so the function is concave

Exponential Functions

For $f(x) = e^x$ :

$f'(x) = e^x > 0$ , monotonically increasing
$f''(x) = e^x > 0$ , convex

Logarithmic Functions

For $f(x) = \ln x$ ( $x > 0$ ):

$f'(x) = \frac{1}{x} > 0$ , monotonically increasing
$f''(x) = -\frac{1}{x^2} < 0$ , concave

Power Functions

For $f(x) = x^n$ :

When $n > 1$ $n > 1$ , $f''(x) = n(n-1)x^{n-2}$ $f^{''} (x) = n (n - 1) x^{n - 2}$
- For $x > 0$ , $f''(x) > 0$ , convex
- For $x < 0$ and $n$ even, $f''(x) > 0$ , convex
- For $x < 0$ and $n$ odd, $f''(x) < 0$ , concave

Applications and Examples

Example 1: Determine the convexity of $f(x) = x^3 - 3x^2 + 2$ .

Solution:

$f'(x) = 3x^2 - 6x$
$f''(x) = 6x - 6 = 6(x-1)$
When $x < 1$ , $f''(x) < 0$ , so the function is concave
When $x > 1$ , $f''(x) > 0$ , so the function is convex

Example 2: Determine the convexity of $f(x) = \frac{1}{x}$ ( $x \neq 0$ ).

Solution:

$f'(x) = -\frac{1}{x^2}$
$f''(x) = \frac{2}{x^3}$
When $x < 0$ , $f''(x) < 0$ , so the function is concave
When $x > 0$ , $f''(x) > 0$ , so the function is convex

Jensen’s Inequality

For a convex function $f$ and weights $\lambda_i \geq 0$ with $\sum_i \lambda_i = 1$ :

f\left(\sum_{i=1}^{n} \lambda_i x_i\right) \leq \sum_{i=1}^{n} \lambda_i f(x_i)

For a concave function, the inequality is reversed:

f\left(\sum_{i=1}^{n} \lambda_i x_i\right) \geq \sum_{i=1}^{n} \lambda_i f(x_i)

Exercises

Exercise 1

Determine the convexity of $f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1$ .

Reference Answer

convexitysecond derivativepolynomial

Approach:

Differentiate twice and analyze the sign.

Steps:

$f'(x) = 4x^3 - 12x^2 + 12x - 4$
$f''(x) = 12x^2 - 24x + 12 = 12(x^2 - 2x + 1) = 12(x-1)^2$
For all $x \neq 1$ , $f''(x) > 0$ , so the function is convex
At $x = 1$ , $f''(1) = 0$ , which needs further analysis

Answer:

The function is convex on both $(-\infty, 1)$ and $(1, +\infty)$ .

Exercise 2

Determine the convexity of $f(x) = \sin x$ .

Reference Answer

convexitytrigonometric functionsecond derivative

Approach:

Differentiate twice and analyze the sign.

Steps:

$f'(x) = \cos x$
$f''(x) = -\sin x$
When $\sin x > 0$ (i.e. $x \in (2k\pi, (2k+1)\pi)$ ), $f''(x) < 0$ , so the function is concave
When $\sin x < 0$ (i.e. $x \in ((2k-1)\pi, 2k\pi)$ ), $f''(x) > 0$ , so the function is convex

Answer:

The function is concave on $(2k\pi, (2k+1)\pi)$ and convex on $((2k-1)\pi, 2k\pi)$ , where $k$ is an integer.

Exercise 3

Determine the convexity of $f(x) = e^{-x^2}$ .

Reference Answer

convexityexponential functionsecond derivative

Approach:

Differentiate twice and analyze the sign.

Steps:

$f'(x) = -2x e^{-x^2}$
$f''(x) = -2 e^{-x^2} + 4x^2 e^{-x^2} = 2e^{-x^2}(2x^2 - 1)$
When $2x^2 - 1 > 0$ (i.e. $|x| > \frac{1}{\sqrt{2}}$ ), $f''(x) > 0$ , so the function is convex
When $2x^2 - 1 < 0$ (i.e. $|x| < \frac{1}{\sqrt{2}}$ ), $f''(x) < 0$ , so the function is concave

Answer:

The function is convex on $(-\infty, -\frac{1}{\sqrt{2}})$ and $(\frac{1}{\sqrt{2}}, +\infty)$ , and concave on $(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$ .

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$f(x)$	Mathematical symbol	f of x	A function of variable $x$
$x_1, x_2$	Mathematical symbol	x one, x two	Two points inside the interval
$\lambda$	Greek letter	Lambda	A parameter with $\lambda \in [0, 1]$
$f'(x)$	Mathematical symbol	f prime of x	The first derivative
$f''(x)$	Mathematical symbol	f double prime of x	The second derivative
$I$	Mathematical symbol	I	An interval

Determination of Extrema

Function Inflection Points

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Convexity and Concavity of Functions

Basic Idea of Convexity

Concave function

Convex function

Concave and Convex Functions

Concave Function

Convex Function

Tests for Convexity

Second-Derivative Test

First-Derivative Test

Convexity of Common Functions

Quadratic Functions

Exponential Functions

Logarithmic Functions

Power Functions

Applications and Examples

Jensen’s Inequality

Exercises

Exercise 1

Exercise 2

Exercise 3

Summary

Symbols Used in This Article

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