What Is a Critical Number Calculator?
A critical number calculator finds the critical numbers of a function — the input values where the derivative is zero or undefined. Critical numbers are the candidates for local maxima, minima, and other key points, making them central to optimization and curve sketching in calculus. Enter a function and the calculator returns its critical numbers with steps.
How to Use the Calculator
- Enter a function f(x).
- Calculate — see the critical numbers and the working.
What Critical Numbers Are
A critical number of a function f is a value x = c in the domain where either:
- f'(c) = 0 — the derivative is zero (horizontal tangent), or
- f'(c) is undefined — the derivative does not exist.
These are the only places where local maxima and minima can occur for a differentiable function.
How to Find Them
| Step | Action |
|---|---|
| 1 | Find the derivative f'(x) |
| 2 | Set f'(x) = 0 and solve |
| 3 | Find where f'(x) is undefined |
| 4 | Keep only values in the domain of f |
Worked Example
For f(x) = x³ − 3x: the derivative is f'(x) = 3x² − 3. Setting it to zero gives x² = 1, so x = 1 and x = −1. These are the critical numbers, where the function has a local maximum and minimum.
Why Critical Numbers Matter
- Optimization: maxima and minima occur at critical numbers.
- Curve sketching: they mark turning points.
- Analysis: combined with the first or second derivative test to classify points.
Frequently Asked Questions
What is a critical number?
A critical number is a value in a function's domain where the derivative is zero or undefined — a candidate for a local maximum or minimum.
How do you find critical numbers?
Take the derivative, set it equal to zero and solve, then also find where the derivative is undefined, keeping only values within the function's domain.
Are all critical numbers maxima or minima?
No — critical numbers are candidates. You must test each with the first or second derivative test to confirm whether it is a max, min, or neither.
What is the difference between a critical number and a critical point?
A critical number is the x-value, while a critical point is the full coordinate (x, f(x)) on the graph at that value.
Is this critical number calculator free?
Yes — it is completely free, requires no signup, and shows the steps to the critical numbers.