In which of the following functions Rolle's theorem is...

In which of the following functions Rolle's theorem is applicable - (A) $f(x)= \begin{cases}x & , 0 \leq x<1 \\ 0 & , x=1\end{cases}$ (B) $f(x)= \begin{cases}\frac{\sin x}{x},-\pi \leq x<0 \\ 0, & x=0\end{cases}$ (C) $f(x)=\frac{x^{2}-x-6}{x-1}$ on $[-2,3]$ (D) $f(x)= \begin{cases}\frac{x^{3}-2 x^{2}-5 x+6}{x-1} & \text { if } x \neq 1, x \in[-2,3] \\ -6 & \text { if } x=1\end{cases}$

In which of the following functions Rolle's theorem is applicable -
(A) $f(x)= \begin{cases}x & , 0 \leq x<1 \\ 0 & , x=1\end{cases}$
(B) $f(x)= \begin{cases}\frac{\sin x}{x},-\pi \leq x<0 \\ 0, & x=0\end{cases}$
(C) $f(x)=\frac{x^{2}-x-6}{x-1}$ on $[-2,3]$
(D) $f(x)= \begin{cases}\frac{x^{3}-2 x^{2}-5 x+6}{x-1} & \text { if } x \neq 1, x \in[-2,3] \\ -6 & \text { if } x=1\end{cases}$

Key Concepts

Rolle's Theorem

Rolle's Theorem is a fundamental result in differential calculus that states if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and takes equal values at the endpoints (i.e., f(a) = f(b)), then there exists at least one point c in (a, b) where the derivative f?'(c) is zero. This theorem guarantees the existence of a stationary point under these specific conditions.

Continuity on a Closed Interval

Continuity on a closed interval [a, b] is a crucial condition for Rolle's Theorem. It means the function does not have any breaks, jumps, or removable discontinuities within and at the boundaries of the interval. Ensuring continuity often requires special attention to piecewise defined functions, especially at points where the function’s rule changes.

Differentiability on an Open Interval

Differentiability on an open interval (a, b) requires that the function has a defined derivative at every point within the interval. This means not only must the function be continuous, but it also must have no sharp corners or cusps in the interval. For piecewise functions or functions with potential removable discontinuities, verifying differentiability can involve simplifying the function to check for smoothness.

Endpoint Equality Condition

The condition f(a) = f(b) is essential for applying Rolle's Theorem. This equality ensures that the function starts and ends at the same value on the interval, which is a necessary precondition for the existence of at least one stationary point where the derivative is zero between a and b.

Removable Discontinuity

A removable discontinuity occurs when a function is undefined at a point or is defined differently from the limit at that point, despite the existence of a well-defined limit from both sides. In the context of Rolle's Theorem, if a function has a removable discontinuity, it may be possible to redefine the function at that point to make it continuous, thus potentially satisfying the theorem’s conditions if the rest of the criteria are met.

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