Critical points calculator
Find all critical points of a function and classify them as maxima, minima, or inflection points using derivative tests.
f(x) =
What are critical points?
Critical points occur where f'(x) = 0 or f'(x) is undefined. They are candidates for local maxima, minima, or neither (inflection points).
Classification using the second derivative test
f''(c) > 0: Local minimum at x = c (concave up).
f''(c) < 0: Local maximum at x = c (concave down).
f''(c) = 0: Inconclusive — use first derivative test or higher-order derivatives.
Inflection points
Points where concavity changes (f'' changes sign). These occur where f''(x) = 0 and f'' actually changes sign.
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