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Q: State And Prove Rolles Theorem With Diagram?

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If a real function, continuous on the segment [a,b] and differentiable on the interval (a, b), takes the same values at the ends of the segment [a,b], then there is at least one point on the interval (a,b) at which the derivative of the function is zero.

Proof

If the function on the segment is constant, then the statement is obvious, since the derivative of the function is zero at any point in the interval.

If not, since the values of the function at the boundary points of the segment are equal, then according to the Weierstrass theorem, it takes its largest or smallest value at some point in the interval, that is, it has a local extremum at this point, and according to Fermat's lemma, the derivative at this point is 0.

Geometric meaning

The theorem states that if the ordinates of both ends of a smooth curve are equal, then there is a point on the curve at which the tangent to the curve is parallel to the abscissa axis.

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