Critical Points Calculator (Local Maxima, Minima and Inflection Points)

How it works

Critical points: f′(x) = 0 · second-derivative test: f″(c) > 0 ⇒ local minimum, f″(c) < 0 ⇒ local maximum, f″(c) = 0 ⇒ check the sign change of f′ · inflection points: f″(x) = 0 with a sign change

A critical point of a differentiable function is an x where the derivative is zero (the tangent line is horizontal). The second derivative tells you the shape there: concave up means a valley (local minimum), concave down a peak (local maximum). If f″ is also zero the first-derivative test decides, and a point where f′ does not change sign is neither. Inflection points are where f″ changes sign. The derivatives are exact; the zeros are found numerically.

Worked example

f(x) = x³ − 3x² − 9x + 5: f′(x) = 3x² − 6x − 9 = 3(x + 1)(x − 3), so the critical points are x = −1, 3. f″(x) = 6x − 6 gives f″(−1) = −12 < 0, a local maximum at (−1, 10), and f″(3) = 12 > 0, a local minimum at (3, −22). f″ = 0 at x = 1, so the inflection point is (1, −6).

Sources: OpenStax Calculus Volume 1 — 4.3 Maxima and Minima · OpenStax Calculus Volume 1 — 4.5 Derivatives and the Shape of a Graph

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