{"id":8155,"date":"2021-11-14T17:54:26","date_gmt":"2021-11-14T12:24:26","guid":{"rendered":"https:\/\/mathemerize.com\/?p=8155"},"modified":"2021-11-15T16:04:00","modified_gmt":"2021-11-15T10:34:00","slug":"verify-rolles-theorem-for-the-function-fx-x2-5x-6-on-the-interval-2-3","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/verify-rolles-theorem-for-the-function-fx-x2-5x-6-on-the-interval-2-3\/","title":{"rendered":"Verify Rolle’s theorem for the function f(x) = \\(x^2\\) – 5x + 6 on the interval [2, 3]."},"content":{"rendered":"

Solution :<\/h2>\n

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

Similar Questions<\/h3>\n

Find the approximate value of f(3.02), where f(x) = \\(3x^2 + 5x + 3\\).<\/a><\/p>\n

If the radius of a sphere is measured as 9 cm with an error of 0.03 cm, then find the approximating error in calculating its volume.<\/a><\/p>\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","protected":false},"excerpt":{"rendered":"

Solution : Since a polynomial function is everywhere differentiable and so continuous also. Therefore, f(x) is continuous on [2, 3] and differentiable on (2, 3). Also, f(2) = \\(2^2\\) – 5 \\(\\times\\) 2 + 6 = 0 and f(3) = \\(3^2\\) – 5 \\(\\times\\) 3 + 6 = 0 \\(\\therefore\\) f(2) = f(3) Thus, all …<\/p>\n

Verify Rolle’s theorem for the function f(x) = \\(x^2\\) – 5x + 6 on the interval [2, 3].<\/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":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[66,43],"tags":[],"yoast_head":"\nVerify Rolle's theorem for the function f(x) = \\(x^2\\) - 5x + 6 on the interval [2, 3]. - Mathemerize<\/title>\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\/verify-rolles-theorem-for-the-function-fx-x2-5x-6-on-the-interval-2-3\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Verify Rolle's theorem for the function f(x) = \\(x^2\\) - 5x + 6 on the interval [2, 3]. - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Solution : Since a polynomial function is everywhere differentiable and so continuous also. Therefore, f(x) is continuous on [2, 3] and differentiable on (2, 3). Also, f(2) = (2^2) – 5 (times) 2 + 6 = 0 and f(3) = (3^2) – 5 (times) 3 + 6 = 0 (therefore) f(2) = f(3) Thus, all … Verify Rolle’s theorem for the function f(x) = (x^2) – 5x + 6 on the interval [2, 3]. 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