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A function f : A->R is given to be differentiable

Math

A function f : A->R is given to be differentiable. A is an open interval.

Let a1 and a2 (a1 < a2) are two real no. in A s.t. f'(a1 ) is not equal to f'(a2).

If a0 is any no. between f'(a1) and f'(a2) then

(i) prove that there exist some x0 in (a1 ,a2 ) s.t. f(x0)=a0.

(iii) Can we conclude that derivative of f has to be continuous on interval A.

Note: This is complete question, no info missing..concepts which may be used to solve this question: Convergence of sequences, continuity of a function, differentiability of a function, Intermediate value theorem, considering cases to prove a thing.

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