{
  "id": "absolute-extrema",
  "version": "fb56c220e0ab",
  "status": "published",
  "name": "Absolute Extrema Calculator",
  "question": "What are the absolute max and min?",
  "summary": "Finds the absolute maximum and minimum of a continuous function on a closed interval [a, b], from its critical numbers and end points.",
  "category": "math",
  "subcategory": "calculus",
  "url": "https://www.acalculator.org/math/absolute-extrema-calculator",
  "markdown": "https://www.acalculator.org/math/absolute-extrema-calculator.md",
  "kind": "function",
  "method": "Closed interval method: f at a, at b, and at each critical number in (a, b), where f′ is 0 or does not exist; the largest is the absolute maximum, the smallest the absolute minimum.",
  "assumptions": [
    "f is continuous on [a, b]; angles in radians.",
    "Values within 10⁻¹² of the extreme value count as ties."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "f": {
        "title": "Function f(x)",
        "description": "A function continuous on [a, b].",
        "type": "string",
        "maxLength": 200
      },
      "a": {
        "title": "From x = a",
        "description": "From x = a: a number or a constant such as pi.",
        "type": "string",
        "maxLength": 40
      },
      "b": {
        "title": "To x = b",
        "description": "To x = b: a number or a constant such as pi.",
        "type": "string",
        "maxLength": 40
      }
    }
  },
  "outputs": {
    "max": {
      "label": "Absolute maximum",
      "description": "The largest value of f on [a, b].",
      "format": "number"
    },
    "maxAt": {
      "label": "Maximum at x =",
      "description": "Where f takes its largest value.",
      "format": "text"
    },
    "min": {
      "label": "Absolute minimum",
      "description": "The smallest value of f on [a, b].",
      "format": "number"
    },
    "minAt": {
      "label": "Minimum at x =",
      "description": "Where f takes its smallest value.",
      "format": "text"
    },
    "candidates": {
      "label": "Values checked",
      "description": "f at each end point and critical number in between.",
      "format": "text"
    },
    "derivative": {
      "label": "f′(x) =",
      "description": "The derivative of f.",
      "format": "math"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "f": "-x^2 + 3x - 2",
      "a": "1",
      "b": "3"
    },
    "outputs": {
      "max": 0.25,
      "maxAt": "3/2",
      "min": -2,
      "minAt": "3",
      "candidates": "f(1) = 0; f(3/2) = 1/4; f(3) = -2",
      "derivative": "3 - 2x"
    },
    "text": "On [1, 3], -x^2 + 3x - 2 has absolute maximum 0.25 at x = 3/2 and absolute minimum -2 at x = 3."
  },
  "examples": [
    {
      "given": {
        "f": "-x^2 + 3x - 2",
        "a": "1",
        "b": "3"
      },
      "expect": {
        "max": 0.25,
        "maxAt": "3/2",
        "min": -2,
        "minAt": "3",
        "candidates": "f(1) = 0; f(3/2) = 1/4; f(3) = -2"
      },
      "source": "OpenStax, Calculus Volume 1, section 4.3 Maxima and Minima, Example 4.13 (https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima), part a"
    },
    {
      "given": {
        "f": "x^2 - 3x^(2/3)",
        "a": "0",
        "b": "2"
      },
      "expect": {
        "max": 0,
        "maxAt": "0",
        "min": -2,
        "minAt": "1"
      },
      "source": "OpenStax, Calculus Volume 1, section 4.3 Maxima and Minima, Example 4.13 (https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima), part b"
    },
    {
      "given": {
        "f": "x^3 - 3x",
        "a": "-2",
        "b": "2"
      },
      "expect": {
        "max": 2,
        "maxAt": "-1, 2",
        "min": -2,
        "minAt": "-2, 1"
      },
      "source": "hand calculation in content.mdx"
    }
  ],
  "sources": [
    "OpenStax, Calculus Volume 1, section 4.3 Maxima and Minima (retrieved 2026-10-03): https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima"
  ],
  "related": [
    "critical-number",
    "inflection-point",
    "derivative",
    "mean-value-theorem"
  ],
  "changelog": []
}