# Cauchys mean value theorem suppose that f and g are

• Notes
• 100000464160110_ch
• 2

This preview shows page 2. Sign up to view the full content.

Cauchy’s Mean Value Theorem. Suppose that f and g are continuous on [ a, b ] and differentiable on ( a, b ). Suppose further that g 0 is never zero on ( a, b ). Then there is some c ( a, b ) such that f ( a ) - f ( b ) g ( a ) - g ( b ) = f 0 ( c ) g 0 ( c ) . Proof. Note first of all that g ( a ) - g ( b ) 6 = 0. Indeed, if g ( a ) = g ( b ) then, by Rolle’s theorem, g 0 ( x ) = 0 at some point x ( a, b ). Let ϕ ( x ) = ( g ( b ) - g ( a )) f ( x ) - ( f ( b ) - f ( a )) g ( x ) + f ( b ) g ( a ) - f ( a ) g ( b ) . Then ϕ ( a ) = ϕ ( b ) = 0 and ϕ satisfies the conditions of Rolle’s theorem. Thus there exists c ( a, b ) such that ϕ 0 ( c ) = ( g ( b ) - g ( a )) f 0 ( c ) - ( f ( b ) - f ( a )) g 0 ( c ) = 0 . This implies the required result. 2
This is the end of the preview. Sign up to access the rest of the document.
• Fall '09
• Calculus, Rolle, local maximum, Rolle's theorem

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern