{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0KN2V7840E"
    }
  ],
  "body": "\n## Statement\n\nLet $I$ be an interval and let $f \\colon I \\to \\mathbb{R}$ be continuous, and differentiable at each interior point of $I$.\nIf $f'(x) \u003e 0$ for each interior point $x$ of $I$, then $f$ is [[strictly increasing|0TR9SR9FTB]] on $I$.\n\n## Remarks\n\nFor $x \u003c y$ in $I$, the [[mean value theorem|0BARQY52N5]] gives $f(y) - f(x) = f'(c)(y - x) \u003e 0$.\nThe converse fails: $x^3$ is strictly increasing, but its derivative at $0$ is $0$.\nMathlib states it as `strictMonoOn_of_deriv_pos`.\n\n## Explanations\n\n### L1 human\n\nIf the graph has positive slope everywhere, it rises from left to right.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "kind": "statement",
    "msc": [
      "26A24"
    ],
    "name": "A function with positive derivative is strictly increasing",
    "role": "corollary",
    "sources": [
      {
        "locator": "Some basic applications and results",
        "url": "https://en.wikipedia.org/wiki/Monotonic_function"
      }
    ],
    "tag": "07J5X2T1ZX",
    "uses": [
      "0BARQY52N5",
      "0TR9SR9FTB"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/07J5X2T1ZX/",
  "links": [
    {
      "field": "uses",
      "target": "0BARQY52N5"
    },
    {
      "field": "uses",
      "target": "0TR9SR9FTB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/07J5X2T1ZX.md",
  "msc": [
    "26A24"
  ],
  "name": "A function with positive derivative is strictly increasing",
  "prov": null,
  "role": "corollary",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/07J/07J5X2T1ZX.md",
  "status": null,
  "tag": "07J5X2T1ZX",
  "uri": "m2://07J5X2T1ZX",
  "url": "https://math2.tamnd.com/wiki/a-function-with-positive-derivative-is-strictly-increasing/",
  "writing": {
    "tag": "07J5X2T1ZX",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}