Determine whether Rolle's Theorem can be applied to f on the...

Determine whether Rolle's Theorem can be applied to $f$ on the closed interval $[a, b] .$ If Rolle's Theorem can be applied, find all values of $c$ in the open interval $(a, b)$ such that $f^{\prime}(c)=0 .$ If Rolle's Theorem cannot be applied, explain why not. $f(x)=\frac{x^{2}-1}{x}, \quad[-1,1]$

Key Concepts

Domain Restrictions in Rational Functions

When dealing with rational functions, it is important to examine the domain carefully because the presence of a variable in the denominator can lead to points where the function is not defined. Such domain restrictions can cause discontinuities or non-differentiability within an interval, thereby preventing the application of theorems like Rolle's.

Differentiability on an Open Interval

Another critical condition for Rolle's Theorem is that the function must be differentiable on the open interval (a, b). Differentiability implies that the function has a well-defined tangent (or derivative) at every point in that interval, allowing one to discuss properties of the derivative such as it being zero at some point.

Continuity on a Closed Interval

A key requirement for applying Rolle's Theorem is that the function must be continuous on the closed interval [a, b]. Continuity ensures that there are no breaks, jumps, or points of undefined behavior within the interval, which is crucial for many existence arguments in analysis.

Rolle's Theorem

Rolle's Theorem is a fundamental result in calculus which states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and satisfies f(a) = f(b), then there exists at least one point c in (a, b) where the derivative f?(c) is zero. This theorem is used to guarantee the existence of stationary points under these specific conditions.

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