Theorem 4.2.5 (Cauchy's mean value theorem). Let f : [a,b] –→R and o: [a,b] → R be continuous functions differentiable on (a,b). Then there exists a point c E (a, b) such that (F(b) – f(a)) o'(c) =f'(c)(@(b) – 9(a)).
Theorem 4.2.5 (Cauchy's mean value theorem). Let f : [a,b] –→R and o: [a,b] → R be continuous functions differentiable on (a,b). Then there exists a point c E (a, b) such that (F(b) – f(a)) o'(c) =f'(c)(@(b) – 9(a)).
Linear Algebra: A Modern Introduction
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Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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Prove the theorem
![Theorem 4.2.5 (Cauchy's mean value theorem). Let f : [a,b] –→R and o: [a,b] → R be continuous
functions differentiable on (a,b). Then there exists a point c E (a,b) such that
(S(b) – f(a)) o'(c) = f'(c)(@(b) – p(a)).
The mean value theorem has the distinction of being one of the few theorems commonly cited
in court. That is, when police measure the speed of cars by aircraft, or via cameras reading license
plates, they measure the time the car takes to go between two points. The mean value theorem then
says that the car must have somewhere attained the speed you get by dividing the difference in
distance by the difference in time.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F718b1378-40e4-4c32-83bc-211fc46d7de4%2Fbbbb9f40-11c6-421d-a019-586e600cc8d3%2Fpf2vb8w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Theorem 4.2.5 (Cauchy's mean value theorem). Let f : [a,b] –→R and o: [a,b] → R be continuous
functions differentiable on (a,b). Then there exists a point c E (a,b) such that
(S(b) – f(a)) o'(c) = f'(c)(@(b) – p(a)).
The mean value theorem has the distinction of being one of the few theorems commonly cited
in court. That is, when police measure the speed of cars by aircraft, or via cameras reading license
plates, they measure the time the car takes to go between two points. The mean value theorem then
says that the car must have somewhere attained the speed you get by dividing the difference in
distance by the difference in time.
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