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4.1 Rolle's theorem and MVT

4.1 Rolle's theorem and MVT - Then there exists c such that...

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4.2 Rolle’s theorem and the mean value theorem Rolle’s Theorem: Given: 1. continuous on [a,b] 2. differentiable on (a,b) 3. f(a)=f(b) Then: there exists c such that a<c<b and f’(c)=0 Proof: Case1: f(x)=k constant F’(c)=0 for all a<c<b Case2: f(x)>f(a) some x is an element of (a,b) F has a max on [a,b] F(c] is max for a<c<b f’(c)=0 (Fermat) Case3: f(x)<f(a) Mean Value Theorem: Given: 1.f is continuous on [a,b] 2. f is differentiable on (a,b)
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Unformatted text preview: Then: there exists c such that a<c<b and f’(c)=f(b)-f(a)/b-a Proof: let: y(x)=f(a)+(f(b)-f(a)/b-a))(x-a) H(x)= f(x)-y(x) 1. h is continuous on [a,b] 2. h is differentiable on [a,b] 3. h(a)=f(a)-y(a)=0 4. h(b)=f(b)-y(b)=0 5. h(a)=h(b) There exists c such that a<c<b and h’(c)=0 (Rolle’s) H’(c)=f’(c)-y’(c)=0 = f’(c)-[f(b)-f(a)/b-a]=0 F’(c)=[f(b)-f(a)/b-a]=0...
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