About this tool
The Critical Points & Extrema Calculator is a free, in-browser tool that analyses the shape of a curve y = f(x). It finds the critical points — the x-values where the first derivative f′(x) equals zero (horizontal tangents) — by scanning the search interval for sign changes of f′ and refining each with bisection. Both the first and second derivatives are computed numerically, so you can enter any function of x without differentiating it yourself.
Everything is calculated locally in your browser, so nothing you type is uploaded. Enter the function using x as the variable (for example x^3 - 3*x, sin(x), or x^4 - 2*x^2) and a search interval, and the tool lists each critical point with its coordinates (x, f(x)) and its classification. Each critical point is classified with the second-derivative test: f″ > 0 means a local minimum (concave up), f″ < 0 means a local maximum (concave down), and when f″ is near zero it falls back to a first-derivative sign-change test to distinguish an extremum from a saddle/inflection.
The calculator also finds inflection points — where the concavity flips, i.e. f″(x) changes sign — and reports them in the same table sorted by x. This is the standard curve-sketching toolkit from calculus I. For f(x) = x³ − 3x it finds a local maximum at x = −1 (value 2), a local minimum at x = 1 (value −2), and an inflection point at x = 0.