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- Rolle's Theorem: Let $f$ be a real valued function that satisfies the following conditions:
- (i) f is defined and continuous on the closed interval $[a,b]$
- (ii) f is differentiable in the open interval $(a,b)$
- (iii) $f(a)=f(b)$
- Then there exists at least one value $c \in (a,b)$ such that $f'(c)=0$

$f(x)=x^2 \qquad 0 \leq x \leq 1$

$f(x)$ being a polynomial function . It is continuous in [-1,1], differentiable in (-1,1)

$f(0)=0 \qquad f(1)=1$

The conditions for Rolle's theorem are not satisfied.

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