Critical Points Calculator

Q: What is a critical point of a function?

A critical point of f(x) is any x-value in the domain where the first derivative f'(x) equals zero or does not exist. These are the candidate locations for local maxima, local minima, and horizontal inflection points, because the tangent line is horizontal (or undefined) there.

Q: How do you find critical points?

First differentiate the function to get f'(x). Then solve f'(x) = 0 and note any points where f'(x) is undefined. Each solution that lies in the domain of f is a critical point. This calculator does it numerically by scanning a range for sign changes of f'(x).

Q: How do you classify a critical point as a max or min?

Use the second-derivative test. Evaluate f''(x) at the critical point: if f'' > 0 the point is a local minimum, if f'' < 0 it is a local maximum, and if f'' = 0 the test is inconclusive (often an inflection or saddle), so you check the sign of f'(x) on each side instead.

Q: What is the difference between a critical point and an inflection point?

A critical point is where f'(x) = 0 or is undefined — it is about the slope of the curve. An inflection point is where the concavity changes, i.e. where f''(x) changes sign. A point can be both, such as the origin for f(x) = x^3, where the slope is zero and the concavity flips.

Q: Does this calculator find every critical point?

It finds every critical point inside the range you specify by densely sampling f'(x) and detecting sign changes, then refining with bisection. Points outside the range, or roots where f'(x) only touches zero without changing sign, may be missed — widen the range or adjust it if you expect more points.

Find the critical points of any function f(x) and classify each as a local minimum, maximum, or inflection point. Enter a function, see f'(x), and get the points instantly.

Updated 2026-06-14 · Free · No sign-up · Runs privately in your browser

Function f(x) Range from Range to Variable

First derivative f'(x)

Second derivative f''(x)

How the points are found

How the Critical Points Calculator Works

This calculator finds the critical points of a function f(x) — the x-values where the slope is flat — and tells you whether each one is a local minimum, local maximum, or an inflection / saddle point. Type any function such as x^3 - 3*x, sin(x), or x^4 - 4*x^2 + 1, set the search range, and the result updates instantly.

Under the hood it differentiates your function symbolically to get f’(x) and f”(x), then scans the range for places where f’(x) crosses zero.

The Method

A critical point occurs where f’(x) = 0 (or f’(x) is undefined). Classify it with the second-derivative test: f”(x) > 0 → minimum, f”(x) < 0 → maximum, f”(x) = 0 → inconclusive.

The tool works numerically:

Differentiate f(x) to obtain f’(x), and again for f”(x).
Sample f’(x) at 2,000 evenly spaced points across your range.
Bracket every interval where f’(x) changes sign — a root of f’(x) = 0 must lie between them.
Bisect each bracket to pin the root to high precision.
Classify each root by the sign of f”(x) at that point.

Worked Example

Take f(x) = x³ − 3x on the range −10 to 10.

f’(x) = 3x² − 3, which is zero when x² = 1, i.e. x = −1 and x = +1.
f”(x) = 6x.
At x = −1: f”(−1) = −6 < 0 → local maximum, with f(−1) = (−1)³ − 3(−1) = 2.
At x = +1: f”(1) = 6 > 0 → local minimum, with f(1) = 1 − 3 = −2.

So the curve peaks at (−1, 2) and dips at (1, −2).

Reading the Second-Derivative Test

f”(x) at critical point	Classification	What the curve does
Positive (f” > 0)	Local minimum	Concave up — a valley
Negative (f” < 0)	Local maximum	Concave down — a peak
Zero (f” = 0)	Inconclusive	Check sign of f’ on each side; often an inflection or saddle

Tips for Accurate Results

Choose a sensible range. The default −10 to 10 covers most textbook problems, but trigonometric functions repeat forever — narrow the range to isolate the points you care about.
Use multiplication signs. Write 3*x, not 3x, and use ^ for powers.
Functions are supported such as sin, cos, tan, exp, log, sqrt, and abs.
Tangent points may be missed. If f’(x) touches zero without crossing (a repeated root), a pure sign-change scan can skip it; the tool falls back to checking the slope on both sides when f” is near zero.

This is ideal for checking calculus homework, sketching graphs, or quickly locating the peaks and valleys of a model before doing the algebra by hand.

Frequently asked questions

What is a critical point of a function?+

A critical point of f(x) is any x-value in the domain where the first derivative f'(x) equals zero or does not exist. These are the candidate locations for local maxima, local minima, and horizontal inflection points, because the tangent line is horizontal (or undefined) there.

How do you find critical points?+

First differentiate the function to get f'(x). Then solve f'(x) = 0 and note any points where f'(x) is undefined. Each solution that lies in the domain of f is a critical point. This calculator does it numerically by scanning a range for sign changes of f'(x).

How do you classify a critical point as a max or min?+

Use the second-derivative test. Evaluate f''(x) at the critical point: if f'' > 0 the point is a local minimum, if f'' < 0 it is a local maximum, and if f'' = 0 the test is inconclusive (often an inflection or saddle), so you check the sign of f'(x) on each side instead.

What is the difference between a critical point and an inflection point?+

A critical point is where f'(x) = 0 or is undefined — it is about the slope of the curve. An inflection point is where the concavity changes, i.e. where f''(x) changes sign. A point can be both, such as the origin for f(x) = x^3, where the slope is zero and the concavity flips.

Does this calculator find every critical point?+

It finds every critical point inside the range you specify by densely sampling f'(x) and detecting sign changes, then refining with bisection. Points outside the range, or roots where f'(x) only touches zero without changing sign, may be missed — widen the range or adjust it if you expect more points.