Rolle's Theorem - Proof of The Generalized Version

Proof of The Generalized Version

Since the proof for the standard version of Rolle's theorem and the generalization are very similar, we prove the generalization.

The idea of the proof is to argue that if f(a) = f(b), then f must attain either a maximum or a minimum somewhere between a and b, say at c, and the function must change from increasing to decreasing (or the other way around) at c. In particular, if the derivative exists, it must be zero at c.

By assumption, f is continuous on, and by the extreme value theorem attains both its maximum and its minimum in . If these are both attained at the endpoints of, then f is constant on and so the derivative of f is zero at every point in (a,b).

Suppose then that the maximum is obtained at an interior point c of (a,b) (the argument for the minimum is very similar, just consider −f ). We shall examine the above right- and left-hand limits separately.

For a real h such that c + h is in, the value f(c + h) is smaller or equal to f(c) because f attains its maximum at c. Therefore, for every h > 0,

hence

where the limit exists by assumption, it may be minus infinity.

Similarly, for every h < 0, the inequality turns around because the denominator is now negative and we get

hence

where the limit might be plus infinity.

Finally, when the above right- and left-hand limits agree (in particular when f is differentiable), then the derivative of f at c must be zero.

Read more about this topic:  Rolle's Theorem

Famous quotes containing the words proof of the, proof of, proof, generalized and/or version:

    Sculpture and painting are very justly called liberal arts; a lively and strong imagination, together with a just observation, being absolutely necessary to excel in either; which, in my opinion, is by no means the case of music, though called a liberal art, and now in Italy placed even above the other two—a proof of the decline of that country.
    Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

    A short letter to a distant friend is, in my opinion, an insult like that of a slight bow or cursory salutation—a proof of unwillingness to do much, even where there is a necessity of doing something.
    Samuel Johnson (1709–1784)

    If we view our children as stupid, naughty, disturbed, or guilty of their misdeeds, they will learn to behold themselves as foolish, faulty, or shameful specimens of humanity. They will regard us as judges from whom they wish to hide, and they will interpret everything we say as further proof of their unworthiness. If we view them as innocent, or at least merely ignorant, they will gain understanding from their experiences, and they will continue to regard us as wise partners.
    Polly Berrien Berends (20th century)

    One is conscious of no brave and noble earnestness in it, of no generalized passion for intellectual and spiritual adventure, of no organized determination to think things out. What is there is a highly self-conscious and insipid correctness, a bloodless respectability submergence of matter in manner—in brief, what is there is the feeble, uninspiring quality of German painting and English music.
    —H.L. (Henry Lewis)

    I should think that an ordinary copy of the King James version would have been good enough for those Congressmen.
    Calvin Coolidge (1872–1933)