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Rolle’s Theorem Lyrics by Math Genius

Calculus

0

If a real-valued function f is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then such a c exists in the open interval (a, b) that f'(c) = 0

Hp.

f(x) continuous on [a,b]
f(x) is differentiable on (a,b)
f(a) = f(b)

Th.

f'(c) = 0
Graphic proof:
Since the function is continuous in [a,b] the arc it draws on the graphic must raise from a, reach a maximum in a certain point c and then reach the x axis again in b. It's easy to see that the slope of the tangent in c must be 0 since the line is horizontal.
Mathematical proof:

Let's study the increment of the function in a point h.

f(c + h) - f(c) =
f(c + h) - f(c) >= 0 h > 0

Apply the Theorem of sign permanence

lim h->0- [f(c + h) - f(c)]/h
lim h->0+ [f(c + h) - f(c)]/h >= 0 h > 0

Use the definition of differentiability

lim h- > 0 [f(c + h) - f(c)]/h = 0

Use the definition of derivative

f'(c) = 0

Q.E.D.
Credits
Math Genius

Math Genius

About “Rolle’s Theorem” by Math Genius

Read the complete lyrics to "Rolle’s Theorem" by Math Genius from the album "Calculus". On Lyrks you can follow along with the full text, explore the artist's discography, and discover related songs. The track is often categorized under Education, Non-Music.

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Frequently asked questions

"Rolle’s Theorem" is credited to Math Genius. Songwriting credits may include additional writers listed on the release; check the credits section on this page for linked collaborators.

"Rolle’s Theorem" appears on "Calculus".

Visit the Math Genius artist page at /artist/math-genius for biography, popular tracks, and links to more lyric pages.

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