Critical Point Finder
Enter a polynomial function of x and find its critical points, classified as local minimums, local maximums, or neither.
How to use this calculator
Type a polynomial function of x, using ^ for exponents — for example x^3 - 3x^2 + 2. Press Calculate to find every critical point and see whether each one is a local minimum, a local maximum, or neither. Use Reset to start over.
What is a critical point?
A critical point of a function is a value of x where the derivative is zero (for a polynomial, this is the only case that applies). At these points, the function's slope momentarily flattens out — which is where local peaks and valleys occur.
Not every critical point is a peak or a valley. Some are inflection-like points where the curve flattens briefly but keeps heading the same direction, such as x = 0 for f(x) = x³.
How to find and classify critical points
2. Solve f'(x) = 0 to get the candidate x-values.
3. Find f''(x), the second derivative.
4. At each candidate point, check the sign of f''(x): positive means a local minimum, negative means a local maximum, and zero needs a closer look (often neither).
Worked example
f'(x) = 3x² − 6x, which is zero at x = 0 and x = 2.
f''(x) = 6x − 6. At x = 0, f''(0) = −6 (negative), so it's a local maximum. At x = 2, f''(2) = 6 (positive), so it's a local minimum.
Frequently asked questions
What's the difference between a critical point and an extremum?
A critical point is any x-value where the derivative is zero. An extremum (a local max or min) is a critical point where the function actually changes direction — so every extremum is a critical point, but not every critical point is an extremum.
Why does the second derivative decide min vs. max?
The second derivative measures how the slope itself is changing. A positive second derivative means the function curves upward (concave up), which happens at a valley. A negative second derivative means it curves downward (concave down), which happens at a peak.
Can a polynomial have no critical points?
Yes. A linear function like f(x) = 2x + 1 has a constant, nonzero derivative, so it never flattens out and has no critical points.
Does this work for non-polynomial functions?
This tool is built for polynomials. Functions involving trig, exponentials, or logarithms need different differentiation rules that aren't covered here.