Critical Points Calculator

Critical points are the places where a curve momentarily stops rising or falling, the peaks, valleys and level spots that shape the whole graph of a function, and this calculator finds them on any interval you choose and tells you what each one is. Enter a function of x and the range of x-values to search, and it returns every critical point it finds, with the x-value, the function value there, and a classification as a local maximum, a local minimum or an inflection-type point, all updating as you type. A critical point is an x-value where the derivative is zero, meaning the tangent to the curve is horizontal, and these are exactly the candidates for the highest and lowest values a function reaches, which is why they sit at the heart of curve sketching and of optimisation problems across science, engineering, economics and everyday decision-making. To classify each one the calculator uses the second derivative test: where the curve is bending upwards the point is a minimum, where it bends downwards it is a maximum, and where the bending vanishes the result is flagged as inconclusive, often an inflection. The tool works numerically, scanning the interval for places where the derivative changes sign, pinning each down precisely, and then checking the curvature, so it handles any function you type, with the caret for powers and the usual sin, cos, sqrt, exp and ln. That makes it genuinely useful for calculus students learning stationary points and the second derivative test and checking homework, for anyone sketching a graph, and for finding the maximum or minimum of a real quantity described by a formula. Because the results refresh live, you can widen the search range or change the function and immediately see the turning points appear and shift. The method and a worked example are explained clearly below.

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Use x, ^ for powers, * for multiply, and functions like sin, cos, sqrt, exp, ln. Derivatives are computed numerically; very flat functions may need a wider range.