Consider the function for x ∈[-2,3], f(x)=[ (x^3-2 x^2-5 x+6)/(x-1) if x ≠ 1 ⌊-6

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Consider the function for x ∈[-2,3], f(x)=[ (x^3-2 x^2-5...

Consider the function for $\mathrm{x} \in[-2,3]$, $f(x)=\left[\begin{array}{ll}\frac{x^{3}-2 x^{2}-5 x+6}{x-1} & \text { if } x \neq 1 \\ \lfloor-6 & \text { if } x=1\end{array}\right.$ then (A) $\mathrm{f}$ is discontinuous at $\mathrm{x}=1 \Rightarrow$ Rolle's theorem is not applicable in $[-2,3]$ (B) $f(-2) \neq f(3) \Rightarrow$ Rolle's theorem is not applicable in $[-2,3]$ (C) $\mathrm{f}$ is not derivable in $(-2,3) \Rightarrow$ Rolle's theorem is not applicable (D) Rolle's theorem is applicable as f satisfies all the conditions and c of Rolle's theorem is $1 / 2$

Consider the function for $\mathrm{x} \in[-2,3]$, $f(x)=\left[\begin{array}{ll}\frac{x^{3}-2 x^{2}-5 x+6}{x-1} & \text { if } x \neq 1 \\ \lfloor-6 & \text { if } x=1\end{array}\right.$ then
(A) $\mathrm{f}$ is discontinuous at $\mathrm{x}=1 \Rightarrow$ Rolle's theorem is not applicable in $[-2,3]$
(B) $f(-2) \neq f(3) \Rightarrow$ Rolle's theorem is not applicable in $[-2,3]$
(C) $\mathrm{f}$ is not derivable in $(-2,3) \Rightarrow$ Rolle's theorem is not applicable
(D) Rolle's theorem is applicable as f satisfies all the conditions and c of Rolle's theorem is $1 / 2$

Key Concepts

Removable Discontinuity

A removable discontinuity occurs at a point where the limit of the function exists, but the function is either not defined at that point or is defined differently from the limit. Although the overall behavior of the function remains predictable and can potentially be 'fixed' by redefining it at that point, such discontinuities can affect the applicability of theorems that require full continuity on an interval.

Differentiability

Differentiability is the property of a function having a derivative at every point within an interval, implying that the function has a well-defined tangent (or instantaneous rate of change) at those points. Differentiability is a stricter condition than continuity and is essential for applying many results in calculus because it ensures smooth behavior without abrupt changes in direction.

Continuity

Continuity refers to the property of a function whereby small changes in the input yield small changes in the output, ensuring no sudden jumps or holes in the graph of the function. For many theorems in calculus, such as Rolle's theorem, the function must be continuous on the entire closed interval being considered.

Rolle's Theorem

Rolle's theorem is a fundamental result in differential calculus which states that 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 highlights the relationship between the function's average rate of change and instantaneous rate of change.

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