{"id":5585,"date":"2021-09-22T01:41:52","date_gmt":"2021-09-21T20:11:52","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5585"},"modified":"2022-01-16T17:03:59","modified_gmt":"2022-01-16T11:33:59","slug":"rolles-theorem","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/rolles-theorem\/","title":{"rendered":"Rolle’s Theorem – Statement and Examples"},"content":{"rendered":"

Here you will learn statement of rolle’s theorem, it’s geometrical and algebraic interpretation with examples.<\/p>\n

Let’s begin –<\/p>\n

Rolle’s Theorem<\/h2>\n
\n

Statement<\/strong> : Let f be a function that satisfies the following three conditions:<\/p>\n

(a) f is continous on the closed interval [a, b].<\/p>\n

(b) f is differentiable on the open interval (a, b)<\/p>\n

(c) f(a) = f(b)<\/p>\n

Then, there exist a real number c \\(\\in\\) (a, b) such that f'(c) = 0.<\/p>\n<\/blockquote>\n

Geometrical Interpretation :<\/strong><\/h4>\n
\n

Geometrically, the theorem says that somewhere between A and B the curve has atleast one tangent parallel to x-axis.<\/p>\n<\/blockquote>\n

Algebraic Interpretation :<\/strong><\/h4>\n
\n

If f is differentiable function then between any two consecutive roots of f(x) = 0, there is atleast one root of the equation f'(x) = 0.<\/p>\n<\/blockquote>\n

Remark – <\/strong>On this theorem generally two types of problems are formulated.<\/p>\n

(a) To check the applicability of rolle’s theorem to a given function on a given interval.<\/p>\n

(b) To verify rolle’s theorem for a given function on a given interval. In both types of problems we first check whether f(x) satisfies conditions of theorem or not. The following results are very helpful in doing so.<\/p>\n

    \n
  1. A polynomial function is everywhere continuous and differentiable.<\/li>\n
  2. The exponential function, sine and cosine functions are everywhere continuous and differentiable.<\/li>\n
  3. Logarithmic function is continuous and differentiable in its domain.<\/li>\n
  4. | x | is not differentiable at x = 0<\/li>\n<\/ol>\n

    Example<\/span><\/strong> : Verify Rolle’s theorem for the function f(x) = \\(x^2\\) – 5x + 6 on the interval [2, 3].<\/p>\n

    Solution<\/strong><\/span> : Since a polynomial function is everywhere differentiable and so continuous also.<\/p>\n

    Therefore, f(x) is continuous on [2, 3] and differentiable on (2, 3).<\/p>\n

    Also, f(2) = \\(2^2\\) – 5 \\(\\times\\) 2 + 6 = 0 and f(3) = \\(3^2\\) – 5 \\(\\times\\) 3 + 6 = 0<\/p>\n

    \\(\\therefore\\) f(2) = f(3)<\/p>\n

    Thus, all the conditions of rolle’s theorem are satisfied. Now, we have to show that there exists some c \\(\\in\\) (2, 3) such that f'(c) = 0.<\/p>\n

    for this we proceed as follows,<\/p>\n

    We have, <\/p>\n

    f(x) = \\(x^2\\) – 5x + 6 \\(\\implies\\) f'(x) = 2x – 5<\/p>\n

    \\(\\therefore\\) f'(x) = 0 \\(\\implies\\) 2x – 5 = 0 \\(\\implies\\) x = 2.5<\/p>\n

    Thus, c = 2.5 \\(\\in\\) (2, 3) such that f'(c) = 0. Hence, Rolle’s theorem is verified.<\/p>\n

    \n

    Related Questions<\/h3>\n

    It is given that for the function f(x) = \\(x^3 \u2013 6x^2 + ax + b\\) on [1, 3], Rolles\u2019s theorem holds with c = \\(2 +{1\\over \\sqrt{3}}\\). Find the values of a and b, if f(1) = f(3) = 0.<\/a><\/p>\n

    Find the point on the curve y = cos x \u2013 1, x \\(\\in\\) \\([{\\pi\\over 2}, {3\\pi\\over 2}]\\) at which tangent is parallel to the x-axis.<\/a><\/p>\n\n\n

    \n
    Next – Lagrange\u2019s Mean Value Theorem<\/a><\/div>\n<\/div>\n\n\n\n
    \n
    Previous – Mean Value Theorems Class 12<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

    Here you will learn statement of rolle’s theorem, it’s geometrical and algebraic interpretation with examples. Let’s begin – Rolle’s Theorem Statement : Let f be a function that satisfies the following three conditions: (a) f is continous on the closed interval [a, b]. (b) f is differentiable on the open interval (a, b) (c) f(a) …<\/p>\n

    Rolle’s Theorem – Statement and Examples<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[37],"tags":[106,108,111,110,109],"yoast_head":"\nRolle's Theorem - Statement and Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn statement of rolle's theorem, it's geometrical and algebraic interpretation with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/rolles-theorem\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Rolle's Theorem - 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