Let f(x)={ x^asin ^2(1)/(n x), x ≠ 0 0 , x=0 . where n ∈I, n ≠0. If Rolle's Th

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Let f(x)={ x^asin ^2(1)/(n x), x ≠ 0 0 ,...

Let $f(x)=\left\{\begin{array}{cl}x^{a} \sin ^{2} \frac{1}{n x}, & x \neq 0 \\ 0 & , x=0\end{array}\right.$ where $n \in I, n \neq 0$. If Rolle's Theorem is applicable to $\mathrm{f}(\mathrm{x})$ in the interval $[0,1]$, then (A) $\alpha>0$, greatest value of $\mathrm{n}$ is $\frac{1}{\pi}$ (B) $\alpha>2$, greatest value of $n$ is $\frac{1}{\pi}$ (C) $\alpha>0$, least value of $\mathrm{n}$ is $-\frac{1}{\pi}$ (D) $\alpha$ can not be $<1$

Let $f(x)=\left\{\begin{array}{cl}x^{a} \sin ^{2} \frac{1}{n x}, & x \neq 0 \\ 0 & , x=0\end{array}\right.$ where $n \in I, n \neq 0$.
If Rolle's Theorem is applicable to $\mathrm{f}(\mathrm{x})$ in the interval $[0,1]$, then
(A) $\alpha>0$, greatest value of $\mathrm{n}$ is $\frac{1}{\pi}$
(B) $\alpha>2$, greatest value of $n$ is $\frac{1}{\pi}$
(C) $\alpha>0$, least value of $\mathrm{n}$ is $-\frac{1}{\pi}$
(D) $\alpha$ can not be $<1$

Key Concepts

Differentiability at a Point with Oscillatory Components

Determining differentiability, especially at a point where the function has an oscillatory term, involves analyzing whether the derivative exists as a well?defined limit. When an oscillatory function is multiplied by a damping factor such as a power of x, the behavior of the damping term must be strong enough to counteract the oscillations for the derivative to exist.

Damping Effects of Power Functions

The role of the power function is to ‘damp’ or control the effect of an oscillatory component near a point (often zero). The exponent in the power function must be chosen so that the product of the power term and the oscillatory term converges appropriately, ensuring both continuity and differentiability at critical points.

Rolle’s Theorem

Rolle’s Theorem is a fundamental result in calculus which states that if a function is continuous on a closed interval, differentiable on the open interval, and takes equal values at the endpoints of that interval, then there exists at least one point in between where the derivative is zero. Its application requires one to check all the stipulated conditions rigorously, especially when the function is defined piecewise.

Continuity of Piecewise-Defined Functions

For a function defined by different expressions on different parts of its domain to be continuous at a point of transition, the limit of the function as the variable approaches that point must equal the function’s value at the point. Establishing continuity is crucial, particularly at the endpoint where the function’s definition changes.

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