How to Find Maxima and Minima of a Function?

Every function that curves up or down has maximum and minimum points — the peaks and valleys of its graph. Finding these points tells you the greatest or least output value a function can produce, whether over its entire domain (absolute extrema) or just in a local region (relative extrema). Algebra 1 focuses especially on quadratic functions, where the vertex gives the maximum or minimum directly.

What Are Maxima and Minima?

• An absolute (global) maximum is the highest output value the function ever reaches.
• An absolute (global) minimum is the lowest output value the function ever reaches.
• A relative (local) maximum is a peak that is higher than all nearby outputs (but not necessarily the overall highest).
• A relative (local) minimum is a valley that is lower than all nearby outputs.

How to Find Maxima and Minima of a Quadratic Function

Using the vertex formula

For \(\color{blue}{f(x) = \text{ ax }}\)² + \(\color{blue}{\text{ bx } + c}\), the vertex x-coordinate is:

Algebra I for Beginners 2027 The Ultimate Step by Step Guide to Acing Algebra I

Download

$27.99 Original price was: $27.99.$17.99Current price is: $17.99.

x = −\(\color{blue}{\frac{b}{(2a)}}\)

Substitute back into f(x) to find the y-coordinate of the vertex.

• If a > 0 (parabola opens upward), the vertex is a minimum.
• If a < 0 (parabola opens downward), the vertex is a maximum.

Example: absolute maximum

f(x) = −x² + \(\color{blue}{6x – 5}\).
a = −1 (opens downward, so there is a maximum).
x = −\(\color{blue}{\frac{6}{(2 \cdot -1)} = 3}\)
f(3) = −\(\color{blue}{9 + 18 – 5}\) = 4
Absolute maximum: 4 at \(\color{blue}{x = 3}\).

Example: absolute minimum

\(\color{blue}{g(x) = x}\)² − \(\color{blue}{4x + 3}\).
\(\color{blue}{a = 1}\) (opens upward, so there is a minimum).
x = −\(\color{blue}{\frac{(-4)}{(2 \cdot 1)} = 2}\)
\(\color{blue}{g(2) = 4 – 8 + 3}\) = −1
Absolute minimum: −1 at \(\color{blue}{x = 2}\).

Reading Maxima and Minima from a Graph

On a graph, a relative maximum is a point that is higher than all points immediately to its left and right. A relative minimum is lower than all points immediately to its left and right. Read the y-value at each peak or valley.

Step-by-Step Summary

1. Identify the type of function (quadratic, etc.).
2. For a quadratic \(\color{blue}{f(x) = \text{ ax }}\)² + \(\color{blue}{\text{ bx } + c}\), find x = −\(\color{blue}{\frac{b}{(2a)}}\).
3. Evaluate f at that x to get the extreme value.
4. Determine whether it is a max (a < 0) or min (a > 0).
5. For non-quadratics, locate peaks and valleys on the graph and read their coordinates.

Watch: Introduction to Maximum and Minimum Points (Video Lesson)

Khan Academy introduces the concept of maximum and minimum points on functions with visual explanations:

Maxima and Minima – Worked Examples

Example 1: Find the maximum of f(x) = −x² + \(\color{blue}{6x – 5}\).

a = −1 < 0 (maximum). X = −\(\color{blue}{\frac{6}{(2 \cdot -1)} = 3}\).
f(3) = −\(\color{blue}{9 + 18 – 5}\) = 4. Maximum value: 4 at \(\color{blue}{x = 3}\).
(Note: \(\color{blue}{f(1) = 0}\) and \(\color{blue}{f(5) = 0}\) are the roots, confirming \(\color{blue}{x = 3}\) is the midpoint/vertex.)

Example 2: Find the minimum of \(\color{blue}{g(x) = x}\)² − \(\color{blue}{4x + 3}\).

\(\color{blue}{a = 1}\) > 0 (minimum). X = −\(\color{blue}{\frac{(-4)}{(2 \cdot 1)} = 2}\).
\(\color{blue}{g(2) = 4 – 8 + 3}\) = −1. Minimum value: −1 at \(\color{blue}{x = 2}\).

Example 3: Find the minimum of \(\color{blue}{f(x) = x}\)² − \(\color{blue}{6x + 8}\).

\(\color{blue}{a = 1}\) > 0. \(\color{blue}{x = \frac{6}{2} = 3}\). \(\color{blue}{f(3) = 9 – 18 + 8}\) = −1. Minimum: −1 at \(\color{blue}{x = 3}\).

Example 4: Find the maximum of h(x) = −2x² + \(\color{blue}{4x + 1}\).

a = −2 < 0. X = −\(\color{blue}{\frac{4}{(2 \cdot -2)} = 1}\). H(1) = −\(\color{blue}{2 + 4 + 1}\) = 3. Maximum: 3 at \(\color{blue}{x = 1}\).

More Practice: Recognizing Relative and Absolute Extrema (Video Lesson)

This second Khan Academy video focuses on distinguishing between relative and absolute maxima and minima from graphs:

Exercises for Maxima and Minima

Find the maximum or minimum value of each function and state the x-value at which it occurs.

1. \(\color{blue}{f(x) = x}\)² − \(\color{blue}{6x + 8}\)
2. f(x) = −x² + \(\color{blue}{4x – 1}\)
3. \(\color{blue}{f(x) = 2x}\)² − \(\color{blue}{8x + 6}\)
4. f(x) = −3x² + \(\color{blue}{6x + 2}\)
5. \(\color{blue}{f(x) = x}\)² + \(\color{blue}{10x + 25}\)

Answers

1. \(\color{blue}{a = 1}\) (min). \(\color{blue}{x = 3}\). \(\color{blue}{f(3) = 9 – 18 + 8}\) = −1 (minimum at \(\color{blue}{x = 3}\))
2. a = −1 (max). \(\color{blue}{x = 2}\). F(2) = −\(\color{blue}{4 + 8 – 1}\) = 3 (maximum at \(\color{blue}{x = 2}\))
3. \(\color{blue}{a = 2}\) (min). \(\color{blue}{x = 2}\). \(\color{blue}{f(2) = 8 – 16 + 6}\) = −2 (minimum at \(\color{blue}{x = 2}\))
4. a = −3 (max). \(\color{blue}{x = 1}\). F(1) = −\(\color{blue}{3 + 6 + 2}\) = 5 (maximum at \(\color{blue}{x = 1}\))
5. \(\color{blue}{a = 1}\) (min). X = −5. \(\color{blue}{f(-5) = 25 – 50 + 25}\) = 0 (minimum at x = −5)

The Ultimate Algebra Bundle 2027 From Pre-Algebra to Algebra II

Download

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.

Want More Practice?

We haven’t published a worksheet built specifically for Maxima and Minima of a Function just yet. In the meantime, the free worksheets below cover closely related skills and concepts. If you’d like extra practice, download any that look helpful, complete the problems, and check your work. They’re a great way to reinforce what you learned on this page and strengthen the foundations this topic builds on:

• Download Characteristics of Quadratic Functions Worksheet
• Download Graphing Quadratic Functions Worksheet

Frequently Asked Questions

How do you find the maximum and minimum of a quadratic function?

The maximum or minimum is at the vertex, where x = −b/(2a). Plug that x back in to get the value. For y = −x² + 4x + 1, x = −4/(−2) = 2 and y = −4 + 8 + 1 = 5. Because a is negative, the parabola opens down, so 5 is the maximum. If a is positive, the vertex is a minimum.

How do you find maxima and minima of a function using derivatives?

Find f′(x), set it equal to 0 and solve to get the critical points. Then test each one. With the second derivative test, f″(x) > 0 means a local minimum and f″(x) < 0 means a local maximum. You can also check whether f′ changes from positive to negative (a maximum) or from negative to positive (a minimum).

What are the maxima and minima of f(x) = 2x³ − 3x² − 36x + 10?

f′(x) = 6x² − 6x − 36 = 6(x − 3)(x + 2), so the critical points are x = 3 and x = −2. f″(x) = 12x − 6 is −30 at x = −2, giving a local maximum f(−2) = 54. It is 30 at x = 3, giving a local minimum f(3) = −71.

What are local maxima and minima of a function?

A local maximum is a point higher than all nearby points, like a hilltop, and a local minimum is lower than all nearby points, like a valley. They do not have to be the highest or lowest values overall. A cubic such as y = x³ − 3x has a local maximum at (−1, 2) and a local minimum at (1, −2), but no absolute extrema.

How do you find the maximum and minimum value of a function on an interval?

For absolute extrema on a closed interval, find the critical points inside the interval, then evaluate the function at those points and at both endpoints. The largest output is the maximum and the smallest is the minimum. For f(x) = x² on [−1, 3], the values are f(−1) = 1, f(0) = 0 and f(3) = 9, so the minimum is 0 and the maximum is 9.

How do you find the maximum and minimum of a sine function?

For y = A sin(Bx) + D, the graph swings |A| units above and below the midline y = D. So the maximum is D + |A| and the minimum is D − |A|. For y = 3 sin x + 1, the maximum is 4 and the minimum is −2. The value of B changes how often the peaks repeat, not how high they are.

Related Topics

• Increasing and Decreasing Functions
• Finding Domain and Range of a Function
• Average Rate of Change of a Function
• Function Notation
• Graphing Rational Functions