{"id":5894,"date":"2025-08-11T10:26:24","date_gmt":"2025-08-11T04:56:24","guid":{"rendered":"https:\/\/study.madeeasy.in\/?p=5894"},"modified":"2025-08-11T10:59:11","modified_gmt":"2025-08-11T05:29:11","slug":"mean-value-theorems","status":"publish","type":"post","link":"https:\/\/www.madeeasy.in\/study\/ee\/calculus-and-differeential-equations\/mean-value-theorems","title":{"rendered":"Mean Value Theorems"},"content":{"rendered":"

Rolle\u2019s Theorem<\/h2>\n

If a function f (x) is such that:\u00a0<\/span><\/p>\n

    \n
  1. f (x) is continuous in the closed interval a \u2264 x \u2264 b and<\/li>\n
  2. f \u2032(x) exists for every point in the open interval a < x < b and<\/li>\n
  3. f(a) = f(b),
    \nThen there exists at least one value of x, say c, where a < c < b such that f\u2032(c) = 0.<\/li>\n<\/ol>\n

    Note:<\/strong> Rolle\u2019s theorem will not hold good.<\/p>\n

      \n
    1. If f(x) is discontinuous at some point in the interval a < x < b<\/li>\n
    2. If f\u2032(x) does not exist at some point in the interval a < x < b or<\/li>\n
    3. If f(a) \u2260 f(b)<\/li>\n<\/ol>\n
      \n
      \n

      Table of Contents<\/p>\nToggle<\/span><\/path><\/svg><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n