📍Critical Points & Extrema Calculator
Enter a polynomial to find its critical points and classify each as a local max, local min, or neither.
Use ^ for powers, terms separated by + or −. Example: 2x^4 - 8x^2 + 3
Free Critical Points Calculator
This critical points calculator finds where a polynomial's derivative equals zero, then applies the first derivative test — checking the sign of f'(x) just before and after each point — to classify it as a local maximum, local minimum, or neither.
Critical points are where a function's slope is momentarily flat, which is exactly where local extrema occur (along with possible inflection points where the slope doesn't change sign). This tool differentiates your polynomial symbolically, solves for its real roots, and evaluates the sign change around each one.
Supports any polynomial degree. Free, runs entirely in your browser, no sign-up required.
About the Critical Points & Extrema Calculator
Features
Any polynomial degree
Works for quadratics, cubics, quartics, and beyond.
Automatic classification
Each critical point is labeled max, min, or neither.
Derivative shown
See f'(x) and how the roots were found.
Instant, private
Runs entirely in your browser. No data is sent anywhere.
Frequently Asked Questions
What is a critical point?
A critical point of f(x) is a value of x where f'(x) = 0 (or is undefined). For polynomials, that means every x-value where the derivative equals zero.
What is the first derivative test?
The first derivative test classifies a critical point by checking the sign of f'(x) just before and after it: positive-to-negative means a local maximum, negative-to-positive means a local minimum, and no sign change means neither.
Are all critical points local extrema?
No. Some critical points are saddle points or inflection points where the slope touches zero but doesn't change sign, so the function keeps increasing or decreasing through that point.
How does this calculator find critical points?
It symbolically differentiates your polynomial term by term, then numerically solves f'(x) = 0 by scanning for sign changes and refining each root with bisection.
What input format does it accept?
Standard polynomial notation using ^ for exponents, such as x^3 - 3x^2 - 9x + 5. Terms can be separated by + or −.
What if my function has no critical points?
Linear functions (degree 1) have a constant, non-zero derivative, so they have no critical points and no local extrema — the calculator will tell you this.
Can it handle high-degree polynomials?
Yes, the numerical root-finding approach works for polynomials of any degree, not just quadratics and cubics that have closed-form solutions.
What's the difference between a critical point and an inflection point?
A critical point is where the first derivative is zero; an inflection point is where the second derivative changes sign (concavity flips). They can occur at different x-values.
Does this find global maximum and minimum?
It finds local extrema. For polynomials with even degree and a positive/negative leading coefficient, the global extremum is typically the most extreme local one, but you should check end behavior for unbounded domains.
Is this calculator free to use?
Yes, completely free with no sign-up, and it runs entirely in your browser.