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

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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) : Conditions of Lagrange's Mean Value Theorem are not satisfied when $f(x)=|x-1|(x-1)$ in $[0,2]$. Reason $(\mathbf{R}):|\mathrm{x}-1|$ is not differentiable at $\mathrm{x}=1$.

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) : Conditions of Lagrange's Mean Value Theorem are not satisfied when $f(x)=|x-1|(x-1)$ in $[0,2]$.
Reason $(\mathbf{R}):|\mathrm{x}-1|$ is not differentiable at $\mathrm{x}=1$.

Key Concepts

Lagrange's Mean Value Theorem

Lagrange's Mean Value Theorem is a fundamental result in calculus that asserts that for any function that is continuous on a closed interval and differentiable on the corresponding open interval, there exists at least one point where the instantaneous rate of change (derivative) equals the average rate of change over the interval. This theorem provides a link between the derivative of a function and its overall behavior and is essential for understanding and proving various results in mathematical analysis.

Continuity and Differentiability Conditions

The theorem's application relies on two essential conditions: the function must be continuous on the closed interval and differentiable on the open interval. Continuity ensures that the function has no breaks or jumps, while differentiability guarantees that the function's derivative exists at every point in the interval (except possibly at the endpoints). Failure to meet either condition invalidates the direct application of the Mean Value Theorem.

Differentiability of Absolute Value Functions

Absolute value functions, due to their piecewise nature, often exhibit points where they are not differentiable, typically at the 'corner' or 'kink' where the definition changes. This non-differentiability at specific points is crucial in analysis because it highlights that while a function might be continuous, the existence of a sharp turn can prevent the function from having a well-defined tangent or derivative at that point.

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