2 8 Rolle s Theorem and the Mean Value Theorem MVT I Rolle s Theorem Let f be continuous on a b and diff able on a b If f a f b then there is at least one c in a b such that f c 0 f a f b a b Examples Find the two x intercepts of intercepts f x x 2 3x 2 and show that f x 0 at some point between the two Work Step 1 0 x 2 0 x x 2 2 0 f 2 0 3x 2 2 x 1 x 1 1 0 f 1 0 Step 2 Realize that f is continuous on 1 2 because it is a polynomial Step 3 And f is diff able on 1 2 because it is a polynomial Step 4 f x 2 x 3 0 2 x 3 3 x 2 3 3 f 0 and 1 2 2 2 See f x x 4 2 x 2 Let Work find all of the values for c in 2 2 such that f c 0 f x 4 x 3 4 x 0 4 x x 2 1 0 4 x x 1 x 1 x 0 1 1 all values for x are in the interval 2 2 II The Mean Value Theorem MVT If in a b such that f c f b f a b a f is continuous on a b and diff able on a b then there exists at least one c represents the slope of the sec line f b f b f a f a b a a b Example f x 5 Work 4 x find all c 1 4 that satisfy the MVT f b f a f 4 f 1 4 1 1 b a 4 1 4 1 Next take the derivative f x 0 4 1 x 2 f x 4 x2 Then set the derivative equal to the slope of the sec line f x 1 4 1 x2 4 x 2 x 2 2 is not in our interval so we have to throw out the negative answer Example f x Work x3 1 c 0 2 4 find all that satisfy the MVT f b f a f 2 f 0 3 1 2 1 b a 2 0 2 0 2 3 f x x2 4 3 2 1 x 4 4 x 2 3 x 2 3 2 3 c 1 15 Toss out negative since it is not in 0 2
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