{
  "id": "critical-number",
  "version": "bc3ef8399d8c",
  "status": "published",
  "name": "Critical Number Calculator",
  "question": "What are the critical numbers of f?",
  "summary": "Finds every critical number of f, where f′ is 0 or does not exist, and which are maxima or minima.",
  "category": "math",
  "subcategory": "calculus",
  "url": "https://www.acalculator.org/math/critical-number-calculator",
  "markdown": "https://www.acalculator.org/math/critical-number-calculator.md",
  "kind": "function",
  "method": "c is a critical number when f(c) is defined and f′(c) is 0 or does not exist. The sign of f′ on each side tells a maximum from a minimum.",
  "assumptions": [
    "x in radians; ln is the natural logarithm.",
    "Points searched for in ±10^6."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "f": {
        "title": "Function f(x)",
        "description": "The function of x.",
        "type": "string",
        "maxLength": 200
      }
    }
  },
  "outputs": {
    "points": {
      "label": "Critical numbers",
      "description": "Where f′ is 0 or does not exist.",
      "format": "text"
    },
    "derivative": {
      "label": "f′(x) =",
      "description": "The derivative of f.",
      "format": "math"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "f": "x^3 - 3x^2 - 9x - 1"
    },
    "outputs": {
      "points": "-1 (local maximum), 3 (local minimum)",
      "derivative": "3x^2 - 6x - 9"
    },
    "text": "The critical numbers of x^3 - 3x^2 - 9x - 1 are -1 (local maximum), 3 (local minimum)."
  },
  "examples": [
    {
      "given": {
        "f": "x^3 - 3x^2 - 9x - 1"
      },
      "expect": {
        "points": "-1 (local maximum), 3 (local minimum)"
      },
      "source": "OpenStax (Strang and Herman, 2016), Calculus Volume 1, section 4.5 Derivatives and the Shape of a Graph, Example 4.17"
    },
    {
      "given": {
        "f": "5x^(1/3) - x^(5/3)"
      },
      "expect": {
        "points": "-1 (local minimum), 0 (f′ does not exist; neither), 1 (local maximum)"
      },
      "source": "OpenStax (Strang and Herman, 2016), Calculus Volume 1, section 4.5 Derivatives and the Shape of a Graph, Example 4.18"
    },
    {
      "given": {
        "f": "x^5 - 5x^3"
      },
      "expect": {
        "points": "-sqrt(3) (local maximum), 0 (neither), sqrt(3) (local minimum)"
      },
      "source": "OpenStax (Strang and Herman, 2016), Calculus Volume 1, section 4.5 Derivatives and the Shape of a Graph, Example 4.20"
    }
  ],
  "sources": [
    "OpenStax, Calculus Volume 1, section 4.3 Maxima and Minima: https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima",
    "OpenStax, Calculus Volume 1, section 4.5 Derivatives and the Shape of a Graph: https://openstax.org/books/calculus-volume-1/pages/4-5-derivatives-and-the-shape-of-a-graph"
  ],
  "related": [
    "tangent-line",
    "instantaneous-rate-of-change",
    "integral",
    "quadratic-formula"
  ],
  "changelog": []
}