Critical Points Calculator

A critical point is a location on a function's graph where the slope goes to zero or does not exist. At that location the function stops increasing and starts decreasing (a local maximum) or stops decreasing and starts increasing (a local minimum), or it may simply flatten out momentarily before continuing in the same direction (a saddle or inflection point). Finding critical points is the foundation of mathematical optimisation: if you want to know the best or worst outcome a process can reach within a given range, you look at the critical points and the boundary values. Engineers use this to minimise cost or maximise efficiency, economists to find equilibrium, and physicists to find stable and unstable equilibria in potential energy surfaces. This calculator scans a function of x over a search interval, identifies every x where the derivative is numerically zero using a sign-change bisection approach, classifies each critical point using the second derivative test, and reports the function value and classification for each one. Enter the function in standard notation, such as x^3 - 3*x, the left and right limits of the search range, and the search resolution. Use ^ for powers, * for multiplication, and Math functions like sin, cos, exp, log, and sqrt. The default example finds critical points of x^3 - 3*x on [-3, 3], yielding a local maximum at x = -1 and a local minimum at x = 1.

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Uses sign-change bisection to find zeros of f'(x), then the second derivative test to classify. Some inflection points may appear if f'' is also near zero.