What is a critical point?
An interior point where f′(x) = 0 or f′ does not exist.
Critical Point Calculator
Enter quadratic or cubic coefficients to locate where the first derivative vanishes.
Enter the polynomial
Choose degree and provide coefficients.
About this calculator
How to use Critical Point Calculator, plus the formulas and logic behind the results.
Tool description
Critical points are candidates for local extrema where f′(x) = 0 or f′ is undefined (polynomials: only f′ = 0).
How to use
- Select quadratic or cubic mode.
- Enter the coefficients.
- Select Calculate to solve f′(x) = 0.
- Evaluate f at each critical x for y-values.
Formulas & reference
Quadratic: f′ = 2ax + b = 0 → x = −b/(2a)Cubic: f′ = 3ax² + 2bx + c = 0 (quadratic formula)
Critical point tips
- Use the second derivative test to classify extrema
- Endpoints of a closed interval are separate candidates
- If f′ has no real root, there is no stationary point
- a = 0 reduces the degree—enter the true leading coefficient
Calculation logic
Differentiates the polynomial and solves the resulting linear or quadratic equation for real roots.
Complex roots of f′ are omitted; only real critical points are listed.
Critical Point Calculator FAQ
Common questions about this calculator and how to use the results.
Quadratic vs cubic?
Quadratics have at most one critical point; cubics can have up to two.
Are critical points always extrema?
No—they can be flat inflection points (e.g. x³ at 0).
What if discriminant of f′ is negative?
No real critical points for that cubic.
Does this include endpoints?
No—only stationary points from f′ = 0.
Can I use decimals?
Yes.