{
  "id": "inflection-point",
  "version": "a6062c1636b6",
  "status": "published",
  "name": "Inflection Point Calculator",
  "question": "Where does the concavity of f change?",
  "summary": "Finds where f is concave up or down, and its inflection points.",
  "category": "math",
  "subcategory": "calculus",
  "url": "https://www.acalculator.org/math/inflection-point-calculator",
  "markdown": "https://www.acalculator.org/math/inflection-point-calculator.md",
  "kind": "function",
  "method": "Concave up where f″ > 0, down where f″ < 0; an inflection point is where f is continuous and f″ changes sign.",
  "assumptions": [
    "x in radians; ln is the natural logarithm.",
    "Points searched for in −10^6 to 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": "Inflection points at x =",
      "description": "Where f is continuous and f″ changes sign.",
      "format": "text"
    },
    "up": {
      "label": "Concave up on",
      "description": "Where f″ > 0.",
      "format": "text"
    },
    "down": {
      "label": "Concave down on",
      "description": "Where f″ < 0.",
      "format": "text"
    },
    "second": {
      "label": "f″(x) =",
      "description": "The second derivative.",
      "format": "math"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "f": "x^3 - 6x^2 + 9x + 30"
    },
    "outputs": {
      "points": "2",
      "up": "(2, ∞)",
      "down": "(-∞, 2)",
      "second": "6x - 12"
    },
    "text": "The inflection points of x^3 - 6x^2 + 9x + 30 are 2."
  },
  "examples": [
    {
      "given": {
        "f": "x^3 - 6x^2 + 9x + 30"
      },
      "expect": {
        "points": "2",
        "up": "(2, ∞)",
        "down": "(-∞, 2)"
      },
      "source": "OpenStax (Strang and Herman, 2016), Calculus Volume 1, section 4.5 Derivatives and the Shape of a Graph, Example 4.19"
    },
    {
      "given": {
        "f": "x^4"
      },
      "expect": {
        "points": "none",
        "up": "(-∞, ∞)",
        "down": "none"
      },
      "source": "OpenStax, Calculus Volume 1, section 4.5, the test for concavity"
    }
  ],
  "sources": [
    "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": [
    "critical-number",
    "tangent-line",
    "curvature",
    "integral"
  ],
  "changelog": []
}