Engineering Calc Tools

Calculus

Critical Points and Extrema Calculator

Critical points, local and absolute extrema, inflection points and where a function increases or is concave up.

Tool no.
ECT-C07
Rev
1.0
Updated
Oct 2026
Method
First and second derivative tests
Examples

Given

Use x as the variable: + − * / ^, sqrt, exp, ln, sin, cos, tan, abs, pi, e. For an optimization problem, type the quantity to maximize as a function of one variable.

Answers

Absolute maximum =—
Runs in your browser. Nothing you enter is uploaded.First and second derivative tests
How to use this calculator

Type f(x) and the interval. Critical points come from f′ = 0 or f′ undefined, inflection points from sign changes of f″, and the absolute extrema from the closed-interval method.

Formulas

Critical pointsf′(c) = 0 or f′(c) undefined
Second-derivative testf″(c) < 0: local max; f″(c) > 0: local min
Closed intervalcompare f at the critical points and at a and b

Worked example

f(x) = x³ − 6x² + 9x + 1 on [−1, 5] (the default)

  1. f′(x) = 3x² − 12x + 9 = 3(x − 1)(x − 3), so x = 1 and x = 3.
  2. f″(x) = 6x − 12: f″(1) = −6 (local max f = 5), f″(3) = 6 (local min f = 1); inflection at x = 2.
  3. Endpoints: f(−1) = −15, f(5) = 21. Absolute max 21 at x = 5, absolute min −15 at x = −1.

Frequently asked questions

How do I use this for optimization?

Write the quantity to optimize as a function of one variable using the constraint, set the physically meaningful interval, and read the absolute extremum.

Why does a point show “neither”?

f′ is zero there but doesn’t change sign, like x³ at 0: a horizontal tangent that isn’t a maximum or minimum.