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Extrema of Functions

Local Maximum

Definition of a Local Maximum

Let $f(x)$ be defined on some neighborhood of $x_0$ . If there exists $\delta > 0$ such that for every $x$ in that neighborhood, $f(x) \leq f(x_0)$

then $f(x_0)$ is called a local maximum of $f(x)$ , and $x_0$ is called a local maximum point.

几何解释

The graph reaches a local highest point at this location.

符号说明

Symbol	Type	Reading/Explanation	Meaning in This Article
$\delta$	Greek letter	Delta	A sufficiently small positive number

Local Minimum

Definition of a Local Minimum

Let $f(x)$ be defined on some neighborhood of $x_0$ . If there exists $\delta > 0$ such that for every $x$ in that neighborhood, $f(x) \geq f(x_0)$

then $f(x_0)$ is called a local minimum of $f(x)$ , and $x_0$ is called a local minimum point.

几何解释

The graph reaches a local lowest point at this location.

Worked Examples

Example 1

Find the extrema of $f(x) = x^2 - 4x + 3$ .

Answer and Explanation(3 个标签)

extremumquadratic functiongraph analysis

Approach: Decide the extrema by graphing and comparing values.

Detailed steps:

Sketch the graph: This is a quadratic function whose graph is an upward-opening parabola.
Identify the extremum point: Since the parabola opens upward, the function has a minimum point and no maximum point.
Find the minimum point: For a quadratic $f(x) = ax^2 + bx + c$ , the axis of symmetry is $x = -\frac{b}{2a}$ $x = -\frac{-4}{2 \times 1} = 2$
Compute the extremum: $f(2) = 2^2 - 4 \times 2 + 3 = 4 - 8 + 3 = -1$
Verify: Take $x = 1$ , $f(1) = 1 - 4 + 3 = 0 > -1$ Take $x = 3$ , $f(3) = 9 - 12 + 3 = 0 > -1$ Confirms $x = 2$ is a minimum point.

Answer: The function attains a local minimum $-1$ at $x = 2$ .

Example 2

Find the extrema of $f(x) = -x^2 + 2x + 1$ .

Answer and Explanation(3 个标签)

extremumquadratic functionmaximum

Approach: Decide the extrema by graphing and comparing values.

Detailed steps:

Sketch the graph: This is a quadratic function whose graph is a downward-opening parabola.
Identify the extremum point: Since the parabola opens downward, the function has a maximum point and no minimum point.
Find the maximum point: For a quadratic $f(x) = ax^2 + bx + c$ , the axis of symmetry is $x = -\frac{b}{2a}$ $x = -\frac{2}{2 \times (-1)} = 1$
Compute the extremum: $f(1) = -1^2 + 2 \times 1 + 1 = -1 + 2 + 1 = 2$
Verify: Take $x = 0$ , $f(0) = 0 + 0 + 1 = 1 < 2$ Take $x = 2$ , $f(2) = -4 + 4 + 1 = 1 < 2$ Confirms $x = 1$ is a maximum point.

Answer: The function attains a local maximum $2$ at $x = 1$ .

Example 3

A rectangle has a fixed perimeter of $20$ cm. Find the maximum possible area.

Answer and Explanation(3 个标签)

extremumapplicationarea maximization

Approach: Build a function relationship and find the extremum by graphing.

Detailed steps:

Build the function: Let the length be $x$ and the width be $y$ . Perimeter: $2(x + y) = 20$ , so $y = 10 - x$ . Area: $S = xy = x(10 - x) = 10x - x^2$ .
Analyze the function: This is a quadratic $S(x) = -x^2 + 10x$ , opening downward, so it has a maximum.
Find the maximum point: Axis of symmetry: $x = -\frac{10}{2 \times (-1)} = 5$ .
Compute the maximum: $S(5) = 10 \times 5 - 5^2 = 50 - 25 = 25$ .
Verify: When $x = 4$ , $S(4) = 40 - 16 = 24 < 25$ . When $x = 6$ , $S(6) = 60 - 36 = 24 < 25$ .

Answer: When the rectangle is a square with sides $5$ cm, the area reaches its maximum of $25$ cm².

Exercises

Exercise 1

Find the extrema of $f(x) = x^2 + 2x + 1$ .

Answer and Explanation(3 个标签)

extremumquadratic functionminimum

Approach: Decide the extrema by graphing and comparing values.

Detailed steps:

Sketch the graph: This is a quadratic function whose graph is an upward-opening parabola.
Identify the extremum point: Since the parabola opens upward, the function has a minimum point and no maximum point.
Find the minimum point: Axis of symmetry: $x = -\frac{b}{2a}$ , so $x = -\frac{2}{2 \times 1} = -1$ .
Compute the extremum: $f(-1) = (-1)^2 + 2 \times (-1) + 1 = 1 - 2 + 1 = 0$ .
Verify: Take $x = -2$ , $f(-2) = 4 - 4 + 1 = 1 > 0$ . Take $x = 0$ , $f(0) = 0 + 0 + 1 = 1 > 0$ . Confirms $x = -1$ is a minimum point.

Answer: The function attains a local minimum $0$ at $x = -1$ .

Exercise 2

Find the extrema of $f(x) = -x^2 + 6x - 5$ .

Answer and Explanation(3 个标签)

extremumquadratic functionmaximum

Approach: Decide the extrema by graphing and comparing values.

Detailed steps:

Sketch the graph: This is a quadratic function whose graph is a downward-opening parabola.
Identify the extremum point: Since the parabola opens downward, the function has a maximum point and no minimum point.
Find the maximum point: Axis of symmetry: $x = -\frac{6}{2 \times (-1)} = 3$ .
Compute the extremum: $f(3) = -3^2 + 6 \times 3 - 5 = -9 + 18 - 5 = 4$ .
Verify: Take $x = 2$ , $f(2) = -4 + 12 - 5 = 3 < 4$ . Take $x = 4$ , $f(4) = -16 + 24 - 5 = 3 < 4$ . Confirms $x = 3$ is a maximum point.

Answer: The function attains a local maximum $4$ at $x = 3$ .

Exercise 3

Adapted from Question 1 of the 2025 National Postgraduate Entrance Exam (Math I)

Is $x=0$ an extremum point of $f(x) = \int_0^x e^{t^2} \sin t\,dt$ ? If so, find the extremum.

Answer and Explanation(3 个标签)

derivativehigher-order derivativeextremum

Approach: This question tests extremum determination for an integral-defined function. Differentiate $f(x)$ , check whether $x=0$ is an extremum point, and compute the extremum.

Detailed steps:

By the fundamental theorem of calculus, $f'(x) = e^{x^2} \sin x$ .
Compute $f'(0)$ : $f'(0) = e^{0} \sin 0 = 1 \times 0 = 0$ , so the derivative vanishes at $x=0$ , which is a possible extremum point.
Differentiate again: $f''(x) = \frac{d}{dx}[e^{x^2} \sin x] = 2x e^{x^2} \sin x + e^{x^2} \cos x$
Compute $f''(0)$ : $f''(0) = 2 \times 0 \times 1 \times 0 + 1 \times 1 = 0 + 1 = 1 > 0$ so $x=0$ is a local minimum point.
Compute the minimum: $f(0) = \int_0^0 e^{t^2} \sin t\,dt = 0$

Answer: $x=0$ is a local minimum point of $f(x)$ , with local minimum value $0$ .

Summary

Symbols Used in This Article

Symbol	Type	Reading/Explanation	Meaning in This Article
$f(x)$	Math symbol	f of x	A function of variable $x$
$x_0$	Math symbol	x zero	The point where the function attains an extremum
$\delta$	Greek letter	Delta	A sufficiently small positive number defining a neighborhood
$f(x_0)$	Math symbol	f of x zero	The function value at $x_0$

English–Chinese Glossary

English term	Chinese term	Phonetic	Explanation
extremum	极值	/ɪkˈstriːməm/	A local maximum or minimum of a function
maximum	极大值	/ˈmæksɪməm/	A local maximum value
minimum	极小值	/ˈmɪnɪməm/	A local minimum value
maximum point	极大值点	/ˈmæksɪməm pɔɪnt/	The point where a local maximum is attained
minimum point	极小值点	/ˈmɪnɪməm pɔɪnt/	The point where a local minimum is attained
extremum point	极值点	/ɪkˈstriːməm pɔɪnt/	A point where a local extremum is attained
neighborhood	邻域	/ˈneɪbəhʊd/	An open interval containing a point
local	局部	/ˈləʊkəl/	Within a neighborhood of a point
parabola	抛物线	/pəˈræbələ/	The graph of a quadratic function
axis of symmetry	对称轴	/ˈæksɪs əv ˈsɪmɪtri/	The symmetry line of a parabola

Parity of Functions

Even and odd functions, their symmetry, key properties, and how to test whether a function is even, odd, or neither.

Determination of Extrema