Assertion (A) and Reason (R) (A) Both A and R are true and R is the correct explanation of A. (B

step 1 So in this question we have been asked if if roll is theorem is not applicable is not applicable

Assertion (A) and Reason (R) (A) Both A and R are true and R...

Assertion (A) and Reason (R) (A) Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$. (B) Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$. (C) $\mathrm{A}$ is true, $\mathrm{R}$ is false. (D) $\mathrm{A}$ is false, $\mathrm{R}$ is true. Assertion $(A):$ If $g(x)$ is a differentiable function $g(2) \neq 0$, $g(-2) \neq 0$ and Rolle's Theorem is not applicable to $f(x)=\frac{x^{2}-4}{g(x)}$ in $[-2,2]$, then $g(x)$ has atleast one root in $(-2,2)$ Reason ( $(\mathrm{R})$ : If a function $\mathrm{f}$ is differentiable in $(\mathrm{a}, \mathrm{b})$ and $f(a)=f(b)$, then Rolle's Theorem is applicable to $f(x)$ for $x \in(a, b)$

Assertion (A) and Reason (R)
(A) Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$.
(B) Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$.
(C) $\mathrm{A}$ is true, $\mathrm{R}$ is false.
(D) $\mathrm{A}$ is false, $\mathrm{R}$ is true.
Assertion $(A):$ If $g(x)$ is a differentiable function $g(2) \neq 0$, $g(-2) \neq 0$ and Rolle's Theorem is not applicable to $f(x)=\frac{x^{2}-4}{g(x)}$ in $[-2,2]$, then $g(x)$ has atleast one root in $(-2,2)$
Reason ( $(\mathrm{R})$ : If a function $\mathrm{f}$ is differentiable in $(\mathrm{a}, \mathrm{b})$ and $f(a)=f(b)$, then Rolle's Theorem is applicable to $f(x)$ for $x \in(a, b)$

Key Concepts

Rolle's Theorem

Rolle's Theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and its values at the endpoints are equal (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 foundational in calculus because it connects the behavior of a function's values to the behavior of its derivative and is applicable only when all its hypotheses, including continuity and differentiability, are strictly met.

Continuity of Rational Functions

For a rational function, such as f(x) = (x² - 4)/g(x), to be continuous over an interval, the denominator must be non-zero throughout that interval. If the denominator g(x) is zero at any point within the interval, the function becomes discontinuous there. This is crucial because many theorems in calculus, including Rolle's Theorem, require the function to be continuous on the closed interval of consideration.

Logical Analysis of Assertion and Reason

In questions involving an assertion and a reason, it is important to independently verify the truth of both statements and determine if the reason correctly explains the assertion. This involves analyzing each statement in the context of the underlying mathematical principles, ensuring that the connection between them is logically sound. This kind of reasoning is essential in abstract mathematical problem solving, where understanding the logical interdependencies is as important as knowing the individual concepts.

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