Unizor - Derivatives - Rolle Theorem

Notes to a video lecture on http://www.unizor.com

Derivatives - Rolle Theorem

Rolle Theorem

If a smooth function f(x), defined on segment [a,b] (including endpoints), has equal values at both endpoints, that is f(a)=f(b), then there exists such point x0 ∈ [a,b] that its derivative at this point f I(x0) equals to zero:
f I(x0) = 0

Proof

Without pretending to be absolutely rigorous, the logical steps to prove this theorem might be as follows.

The function f(x) cannot be monotonically increasing because in this case its value at point x=b would be greater than that at point x=a.

Analogously, the function f(x) cannot be monotonically decreasing because in this case its value at point x=b would be less than that at point x=a.

Therefore, the function either is constant, in which case it's derivative at any internal point in the interval (a,b) equals to zero, or the function changes increasing to decreasing or decreasing to increasing behavior somewhere inside this interval.

Any point of change from increasing to decreasing is a local maximum, any point of change from decreasing to increasing is a local minimum. In both cases, according Fermat's Theorem, a derivative must be equal to zero at a point of change.

End of proof.