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.
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