Absolute Extrema Finder on Interval

Math Calculus • Applications of Derivatives

Written by STEM Calculators Team Published June 15, 2026 Updated June 15, 2026

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Theory

Closed-interval method

The absolute extrema of a function on a closed interval are the largest and smallest function values on that interval. For a continuous function on \([a,b]\), the Extreme Value Theorem guarantees that both an absolute maximum and an absolute minimum exist.

Step 1. Start with the closed interval.

\[ \begin{aligned} x &\in [a,b] \end{aligned} \]

Step 2. Find the interior critical points.

A critical point \(c\) is checked only when it lies inside the interval:

\[ \begin{aligned} a &< c < b \\ f'(c) &= 0 \end{aligned} \]

In many textbook problems, you also check points where \(f'(c)\) is undefined, as long as \(f(c)\) is defined.

Step 3. Build the candidate set.

The candidate set includes the two endpoints and all interior critical points:

\[ \begin{aligned} C &= \{a,b,c_1,c_2,\ldots,c_n\} \end{aligned} \]

Step 4. Evaluate and compare.

Evaluate the function at every candidate. The smallest value is the absolute minimum, and the largest value is the absolute maximum.

\[ \begin{aligned} \text{absolute minimum} &= \min\{f(x):x\in C\} \\ \text{absolute maximum} &= \max\{f(x):x\in C\} \end{aligned} \]

Important: endpoints are not optional. Many absolute maxima or minima occur at endpoints.

Worked example

Find the absolute extrema of \(f(x)=x^3-3x\) on \([-2,3]\).

Step 1. Differentiate the function.

\[ \begin{aligned} f(x) &= x^3-3x \\ f'(x) &= 3x^2-3 \end{aligned} \]

Step 2. Solve \(f'(x)=0\).

\[ \begin{aligned} 3x^2-3 &= 0 \\ 3(x^2-1) &= 0 \\ x^2-1 &= 0 \\ x &= -1,\ 1 \end{aligned} \]

Both critical points are inside the interval \([-2,3]\).

Step 3. Evaluate endpoints and critical points.

\[ \begin{aligned} f(-2) &= (-2)^3-3(-2) = -2 \\ f(-1) &= (-1)^3-3(-1) = 2 \\ f(1) &= 1^3-3(1) = -2 \\ f(3) &= 3^3-3(3) = 18 \end{aligned} \]

Step 4. Compare the candidate values.

\[ \begin{aligned} \text{absolute minimum} &= -2 &&\text{at }x=-2\text{ and }x=1 \\ \text{absolute maximum} &= 18 &&\text{at }x=3 \end{aligned} \]

Common mistakes

Forgetting endpoints: endpoints can be absolute extrema even when they are not critical points.
Checking points outside the interval: only critical points inside \((a,b)\) are used.
Comparing \(x\)-values instead of \(f(x)\)-values: extrema are based on function values.
Assuming every critical point is absolute: critical points are only candidates; comparison decides.

Frequently Asked Questions

How do you find absolute extrema on a closed interval?

Evaluate the function at the endpoints and at every critical point inside the interval. The largest value is the absolute maximum and the smallest value is the absolute minimum.

Why do endpoints matter?

An absolute maximum or minimum on a closed interval can occur at an endpoint, even if the derivative is not zero there.

What is a critical point?

A critical point is an interior point where f'(x)=0 or where f'(x) is undefined while the function itself is defined.

What are the absolute extrema of f(x)=x^3-3x on [-2,3]?

The candidate points are x=-2, x=-1, x=1, and x=3. The absolute minimum is f(-2)=-2 and the absolute maximum is f(3)=18.

Can an absolute maximum and minimum occur at the same point?

Yes. If the function is constant on the interval, every point has the same value, so every point is both an absolute maximum and an absolute minimum.

Does the calculator require the function to be continuous?

The closed-interval theorem guarantees absolute extrema for continuous functions on closed intervals. The calculator can still evaluate candidates numerically, but discontinuities require extra care.

Does the graph show the extrema?

Yes. The graph highlights endpoints, critical points, absolute maximum points, and absolute minimum points.

Can I use trigonometric functions?

Yes. The calculator supports sin, cos, tan, pi, and typed interval endpoints such as 0 and 2pi.

Absolute Extrema Finder on Interval

Function and interval

Graph and output settings

Quick examples

Graph with highlighted absolute extrema

Step-by-step closed-interval method

Rate this calculator

Frequently Asked Questions

Function and interval

Graph and output settings

Quick examples

Graph with highlighted absolute extrema

Step-by-step closed-interval method

Rate this calculator

Frequently Asked Questions

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