Lect24 - Rolle's Theorem Let f be a function that satisfies...

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92.131 Lecture 24 1 of 6 Ronald Brent © 2009 All rights reserved. Rolle’s Theorem L e t f be a function that satisfies the following three hypotheses: 1 ) f is continuous on the closed interval ] , [ b a . 2 ) f is differentiable on the open interval ) , ( b a . 3 ) ) ( ) ( b f a f = Then there is a number c in ) , ( b a such that 0 ) ( = c f .
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92.131 Lecture 24 2 of 6 Ronald Brent © 2009 All rights reserved. Examples x y ab c 1 c 2 x y a b c x y a b c 1 c 2 x y a b c
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92.131 Lecture 24 3 of 6 Ronald Brent © 2009 All rights reserved. The Mean Value Theorem (MVT) Let f be a function that satisfies the following hypotheses: 1 ) f is continuous on the closed interval ] , [ b a . 2 ) f is differentiable on the open interval ) , ( b a . Then there is a number c in ) , ( b a such that a b a f b f c f = ) ( ) ( ) (, or equivalently, ) )( ( ) ( ) ( a b c f a f b f = . ) ( c f is the average (mean) value of the function on the interval ) , ( b a .
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92.131 Lecture 24 4 of 6 Ronald Brent © 2009 All rights reserved. Geometrically: x y a b c x y a b c 1 c 2
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92.131 Lecture 24 5 of 6 Ronald Brent © 2009 All rights reserved.
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Lect24 - Rolle's Theorem Let f be a function that satisfies...

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