Consider f(x)=|1-x| 1<x<2 1 ≤x ≤2 and g(x)=f(x)+b sinπ/ 2 x, 1 ≤x ≤2 then which of the follo

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Consider f(x)=|1-x| 1<x<2 1 ≤x ≤2 and g(x)=f(x)+b sinπ/ 2...

Consider $\mathrm{f}(\mathrm{x})=|1-\mathrm{x}| 1<\mathrm{x}<2 \quad 1 \leq \mathrm{x} \leq 2$ and $g(x)=f(x)+b \sin \pi / 2 x, \quad 1 \leq x \leq 2$ then which of the following is correct? (A) Rolles Theorem is applicable to both $\mathrm{f}, \mathrm{g}$ and $\mathrm{b}=3 / 2$ (B) LMVT is not applicable to $\mathrm{f}$ and Rolle's Theorem if applicable to $g$ with $b=1 / 2$ (C) LMVT is applicable to $\mathrm{f}$ and Rolle's Theorem is applicable to $g$ with $b=1$ (D) Rolle's Theorem is not applicable to both $\mathrm{f}, \mathrm{g}$ for any real $\underline{b}$

Consider $\mathrm{f}(\mathrm{x})=|1-\mathrm{x}| 1<\mathrm{x}<2 \quad 1 \leq \mathrm{x} \leq 2$
and $g(x)=f(x)+b \sin \pi / 2 x, \quad 1 \leq x \leq 2$
then which of the following is correct?
(A) Rolles Theorem is applicable to both $\mathrm{f}, \mathrm{g}$ and $\mathrm{b}=3 / 2$
(B) LMVT is not applicable to $\mathrm{f}$ and Rolle's Theorem if applicable to $g$ with $b=1 / 2$
(C) LMVT is applicable to $\mathrm{f}$ and Rolle's Theorem is applicable to $g$ with $b=1$
(D) Rolle's Theorem is not applicable to both $\mathrm{f}, \mathrm{g}$ for any real $\underline{b}$

Key Concepts

Mean Value Theorem

The Mean Value Theorem (MVT) states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) where the derivative of the function equals the average rate of change over [a, b]. This theorem is fundamental in analysis as it bridges the behavior of a function on an interval to its instantaneous rate of change.

Rolle's Theorem

Rolle's Theorem is a special case of the Mean Value Theorem. It asserts that if a function is continuous on the 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 is frequently used to demonstrate the existence of stationary points within an interval.

Continuity and Differentiability on Closed Intervals

For both the Mean Value Theorem and Rolle's Theorem to be applicable, the function must be continuous on the closed interval and differentiable on the open interval. While continuity ensures no breaks or jumps in the function's graph, differentiability guarantees that the function has a well-defined tangent (and therefore slope) at all interior points. In problems like this, checking these conditions is crucial to determining whether the theorems apply.

Analysis of Piecewise and Transformed Functions

When a function involves an absolute value or is defined in a piecewise manner, it is important to analyze how it behaves on the interval in question. This typically involves 'unwrapping' the absolute value to find an equivalent expression that is easier to work with, and verifying that the necessary conditions (like continuity and differentiability) remain valid. Additionally, when another function is defined by adding a transformation term, any parameter involved must be chosen so that key conditions for the theorems (such as equal endpoint values for Rolle’s Theorem) are satisfied.

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